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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<HTML LANG="EN-US">
<HEAD>
<!-- improve mobile display -->
<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<TITLE>Home - Intuitionistic Logic Explorer</TITLE>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<LINK REL="shortcut icon" HREF="favicon.ico" TYPE="image/x-icon">
<STYLE TYPE="text/css">
<!--
/* Math symbol image will be shifted down 4 pixels to align with normal
text for compatibility with various browsers. The old ALIGN=TOP for
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All other images must override this shift with STYLE="margin-bottom:0px".
(2-Oct-2015 nm) */
img { margin-bottom: -4px }
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</STYLE>
</HEAD>
<!-- <BODY BGCOLOR="#EEFFFA"> -->
<BODY BGCOLOR="#FFFFFF" STYLE="padding: 0px 8px">
<TABLE BORDER=0 CELLSPACING=0 CELLPADDING=0 WIDTH="100%">
<TR>
<TD ALIGN=LEFT VALIGN=TOP><A HREF="../index.html"><IMG SRC="_icon-il.gif"
BORDER=0
ALT="Metamath Home"
TITLE="Metamath Home"
HEIGHT=32 WIDTH=32 ALIGN=TOP STYLE="margin-bottom:0px"></A>
</TD>
<TD ALIGN=CENTER VALIGN=TOP><FONT SIZE="+3"
COLOR="#006633"><B>Intuitionistic Logic Explorer Home Page</B></FONT>
</TD>
<TD NOWRAP ALIGN=RIGHT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif> <A HREF="wn.html">First ></A><BR><A
HREF="ax1hfs.html">Last ></A></FONT>
</TD>
</TR>
<TR>
<TD COLSPAN=3 ALIGN=LEFT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif>
<A HREF="../mm.html">Mirrors</A> >
<A HREF="../index.html">Home</A> >
ILE Home >
<A HREF="mmtheorems.html">Th. List</A> >
<A HREF="mmrecent.html">Recent</A>
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<HR NOSHADE SIZE=1>
<CENTER>
<B><FONT COLOR="#006633">Created by Mario Carneiro</FONT></B>
</CENTER>
<HR NOSHADE SIZE=1>
<B><FONT COLOR="#006633">Intuitionistic Logic Proof Explorer</FONT></B>
<!--
<P><I>Intuitionistic Logic</I> (<A
HREF="https://en.wikipedia.org/wiki/Intuitionistic_logic">Wikipedia</A>
[accessed 19-Jul-2015], <A
HREF="http://plato.stanford.edu/entries/logic-intuitionistic/"> Stanford
Encyclopedia of Philosophy</A> [accessed 19-Jul-2015]) is a logic weaker
than classical logic that can be thought of as a weakening of classical
logic such that the law of excluded middle, (<FONT
COLOR="#0000FF"><I>φ</I></FONT> <FONT FACE=sans-serif> ∨</FONT>
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>), doesn't always hold.
Specifically, it holds if we have a proof for <FONT
COLOR="#0000FF"><I>φ</I></FONT> or we have a proof for ¬ <FONT
COLOR="#0000FF"><I>φ</I></FONT>, but it doesn't necessarily hold if
we don't have a proof of either one. Intuitionistic logic can be
thought of as a constructive logic in which we must build and exhibit
concrete examples of objects before we can accept their existence. A
proof by contradiction, where denial of an assertion to be proved leads
to asserting a falsehood, will generally not be valid in intuitionistic
logic.
-->
<!-- the above was replaced by David A. Wheeler: -->
<P><I>Intuitionistic Logic</I> (<A
HREF="https://en.wikipedia.org/wiki/Intuitionistic_logic">Wikipedia</A>
[accessed 19-Jul-2015], <A
HREF="http://plato.stanford.edu/entries/logic-intuitionistic/"> Stanford
Encyclopedia of Philosophy</A> [accessed 19-Jul-2015]) can be
thought of as a constructive logic in which we must build and exhibit
concrete examples of objects before we can accept their existence.
Unproved statements in intuitionistic logic are not given an intermediate truth value,
instead, they remain of unknown truth value until they are either proved or disproved.
Intuitionist logic can also be thought of as a weakening of classical
logic such that the law of excluded middle (LEM), (<FONT
COLOR="#0000FF"><I>φ</I></FONT> <FONT FACE=sans-serif> ∨</FONT>
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>), doesn't always hold.
Specifically, it holds if we have a proof for <FONT
COLOR="#0000FF"><I>φ</I></FONT> or we have a proof for ¬ <FONT
COLOR="#0000FF"><I>φ</I></FONT>, but it doesn't necessarily hold if
we don't have a proof of either one.
There is also no rule for double negation elimination.
Brouwer observed in 1908 that LEM was abstracted from finite situations,
then extended without justification to statements about infinite collections.
<HR NOSHADE SIZE=1>
<TABLE WIDTH="100%"><TR>
<TD VALIGN=top>
<B><FONT COLOR="#006633">Contents of this page</FONT></B>
<MENU>
<LI> <A HREF="#overview">Overview of intuitionistic logic</A></LI>
<LI> <A HREF="#overview2">Overview of this work</A></LI>
<LI> <A HREF="#axioms">The axioms</A></LI>
<LI> <A HREF="#theorems">Some theorems</A></LI>
<LI> <A HREF="#intuitionize">How to intuitionize classical proofs</A></LI>
<LI> <A HREF="#setmm">Metamath Proof Explorer cross reference</A></LI>
<LI> <A HREF="#bib">Bibliography</A></LI>
</MENU></TD>
<TD VALIGN=top>
<B><FONT COLOR="#006633">Related pages</FONT></B>
<MENU>
<LI> <A HREF="mmtheorems.html">Table of Contents and Theorem List</A></LI>
<LI>
<A HREF="mmrecent.html">Most Recent Proofs
(this mirror)</A>
(<A HREF="http://us2.metamath.org:88/ileuni/mmrecent.html">latest</A>)
</LI>
<LI> <A HREF="mmbiblio.html">Bibliographic Cross-Reference</A></LI>
<LI> <A HREF="mmdefinitions.html">Definition List</A></LI>
<LI> <A HREF="mmascii.html">ASCII Equivalents for Text-Only Browsers</A></LI>
<LI>
<A HREF="../metamath/iset.mm">Metamath database iset.mm (ASCII file)</A>
</LI>
</MENU>
<B><FONT COLOR="#006633">External links</FONT></B>
<MENU>
<LI>
<A HREF=" https://github.com/metamath/set.mm">GitHub repository</A>
[accessed 06-Jan-2018]
</LI>
</MENU>
</TD>
</TR></TABLE>
<HR NOSHADE SIZE=1><A NAME="overview"></A><B><FONT COLOR="#006633">
Overview of intuitionistic logic</FONT></B>
<P>(Placeholder for future use)
<HR NOSHADE SIZE=1><A NAME="overview2"></A><B><FONT COLOR="#006633">
Overview of this work</FONT></B>
<P>(By Gérard Lang, 7-May-2018)
<P>Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a
development of "set.mm" whose eventual
aim is to show how many of the theorems of set theory and
mathematics that can be derived from classical first order logic can
also be derived from a weaker system called "intuitionistic logic." To
achieve this task, iset.mm adds (or substitutes) intuitionistic
axioms for a number of the classical logical axioms of set.mm.
<P>Among these new axioms, the first six
( ~ ax-ia1 ,
~ ax-ia2 ,
~ ax-ia3 ,
~ ax-io ,
~ ax-in1 ,
and
~ ax-in2 ), when added to
~ ax-1 ,
~ ax-2 ,
and
~ ax-mp ,
allow for the development of intuitionistic propositional logic.
We omit the classical axiom
<SPAN CLASS=math>((¬
<SPAN CLASS=wff STYLE="color:blue">𝜑</SPAN> → ¬ <SPAN
CLASS=wff STYLE="color:blue">𝜓</SPAN>) → (<SPAN CLASS=wff
STYLE="color:blue">𝜓</SPAN> → <SPAN CLASS=wff
STYLE="color:blue">𝜑</SPAN>))</SPAN> (which is ax-3 in
set.mm). Each of our new axioms is a theorem of classical
propositional logic, but ax-3 cannot be derived from them. Similarly,
other basic classical theorems, like the third middle excluded or the
equivalence of a proposition with its double negation, cannot be derived
in intuitionistic propositional calculus. Glivenko showed that a
proposition <FONT COLOR="#0000FF"><I>φ</I></FONT>
is a theorem of classical propositional calculus if and only
if ¬¬<FONT COLOR="#0000FF"><I>φ</I></FONT>
is a theorem of intuitionistic propositional calculus.
<P>The next 4 new axioms
( ~ ax-ial ,
~ ax-i5r ,
~ ax-ie1 ,
and
~ ax-ie2 )
together with the set.mm axioms
~ ax-4 ,
~ ax-5 ,
~ ax-7 ,
and
~ ax-gen
do not mention equality or distinct variables.
<P>The ~ ax-i9 axiom is just a slight variation of the classical ~ ax-9 .
The classical axiom ~ ax-12 is strengthened into first ~ ax-i12 and then
~ ax-bndl (two results which would be fairly readily equivalent to ~ ax-12
classically but which do not follow from ~ ax-12 , at least not in an obvious
way, in intuitionistic logic).
The substitution of ~ ax-i9 , ~ ax-i12 , and ~ ax-bndl for ~ ax-9 and ~ ax-12
and the inclusion of
~ ax-8 ,
~ ax-10 ,
~ ax-11 ,
~ ax-13 ,
~ ax-14 ,
and
~ ax-17
allow for the development of the intuitionistic predicate calculus.
<P>Each of the new axioms is a theorem of classical first order
logic with equality. But some axioms of classical first order logic
with equality, like ax-3, cannot be derived in the intuitionistic
predicate calculus.</P>
<P>One of the major interests of the intuitionistic predicate calculus
is that its use can be considered as a realization of the program of the
constructivist philosophical view of mathematics.
<HR NOSHADE SIZE=1><A NAME="axioms"></A><B><FONT COLOR="#006633">
The axioms</FONT></B>
<P>As with the <A HREF="mmset.html#axioms">classical axioms</A>
we have propositional logic and predicate logic.
<P>The axioms of intuitionistic propositional logic consist of some of the axioms from
classical propositional logic, plus additional axioms for the operation of the 'and',
'or' and 'not' connectives.
<CENTER>
<TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Axioms of intuitionistic propositional calculus">
<CAPTION><B>Axioms of intuitionistic propositional calculus</B></CAPTION>
<TR ALIGN=LEFT><TD> <A HREF="ax-1.html"> Axiom <I>Simp</I></A></TD>
<TD><FONT COLOR="#006633"><B>ax-1</B></FONT></TD>
<TD><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → (<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>))</TD>
</TR>
<TR ALIGN=LEFT><TD><A HREF="ax-2.html">Axiom <I>Frege</I></A></TD> <TD
NOWRAP><FONT COLOR="#006633"><B>ax-2</B></FONT></TD>
<TD>
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>((<FONT
COLOR="#0000FF"><I>φ</I></FONT> → (<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
COLOR="#0000FF"><I>χ</I></FONT>)) → ((<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>χ</I></FONT>)))</TD>
</TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-mp.html">Rule of Modus Ponens</A></TD> <TD NOWRAP><FONT
COLOR="#006633"><B>ax-mp</B></FONT></TD>
<TD>` |- ph ` &
` |- ph -> ps ` ⇒ ` |- ps `</TD>
</TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia1.html">Left 'and' elimination</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia1</B></FONT></TD><TD>
` |- ( ( ph /\ ps ) -> ph ) `</TD></TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia2.html">Right 'and' elimination</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia2</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>((<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif">∧</FONT> <FONT COLOR="#0000FF"><I>ψ</I></FONT>) →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>)</TD></TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia3.html">'And' introduction</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia3</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(<FONT COLOR="#0000FF"><I>φ</I></FONT>
→ (<FONT COLOR="#0000FF"><I>ψ</I></FONT> → (<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif">∧</FONT> <FONT COLOR="#0000FF"><I>ψ</I></FONT>)))</TD></TR>
<TR ALIGN="LEFT"><TD><A
HREF="ax-io.html">Definition of 'or'</A></TD><TD><FONT COLOR="#006633"><B>ax-io</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(((<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif"> ∨</FONT> <FONT COLOR="#0000FF"><I>χ</I></FONT>)
→ <FONT COLOR="#0000FF"><I>ψ</I></FONT>)
↔ ((<FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>) <FONT FACE="sans-serif">∧</FONT>
(<FONT COLOR="#0000FF"><I>χ</I></FONT> → <FONT COLOR="#0000FF"><I>ψ</I></FONT>)))
<TR ALIGN="LEFT"><TD><A
HREF="ax-in1.html">'Not' introduction</A></TD><TD><FONT COLOR="#006633"><b>ax-in1</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢
</FONT>((<FONT COLOR="#0000FF"><I>φ</I></FONT> → ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>)
→ ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>)</TD></TR>
<TR ALIGN="LEFT"><TD><A
HREF="ax-in2.html">'Not' elimination</A></TD><TD><FONT COLOR="#006633"><B>ax-in2</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>
→ (<FONT COLOR="#0000FF"><I>φ</I></FONT> → <FONT COLOR="#0000FF"><I>ψ</I></FONT>))</TD></TR>
</TABLE>
</CENTER>
<P>Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of
implication and 'not'. In particular, <FONT COLOR="#0000FF"><I>φ</I></FONT> ∨
<FONT COLOR="#0000FF"><I>ψ</I></FONT> is not equivalent to ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>,
nor is <FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT> equivalent to ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT> ∨
<FONT COLOR="#0000FF"><I>ψ</I></FONT>, nor is <FONT COLOR="#0000FF"><I>φ</I></FONT> ∧
<FONT COLOR="#0000FF"><I>ψ</I></FONT> equivalent to ¬ (<FONT COLOR="#0000FF"><I>φ</I></FONT> → ¬
<FONT COLOR="#0000FF"><I>ψ</I></FONT>).</P>
<P>The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if
<FONT COLOR="#0000FF"><I>φ</I></FONT> implies a contradiction (such as its own negation),
then one can conclude ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>. By contrast, assuming
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>
and then deriving a contradiction only serves to prove ¬ ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>,
which in intuitionistic logic is not the same as <FONT COLOR="#0000FF"><I>φ</I></FONT>.</P>
<P>The biconditional can be defined as the conjunction of two implications, as in
<A HREF="dfbi2.html">dfbi2</a> and <A HREF="df-bi.html">df-bi</a>.</P>
<P><A NAME="pcaxioms"></A><B><FONT COLOR="#006633">Predicate
logic</FONT></B> adds set variables (individual variables) and the ability to quantify
them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃
cannot be defined in terms of ∀. As in classical logic, we also add = for equality
(which is key to how we handle substitution in metamath) and ∈ (which for current
purposes can just be thought of as an arbitrary predicate, but which will later come to
mean set membership).</P>
<p>Our axioms are based on the classical set.mm predicate logic axioms, but adapted for
intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing
some aspects of how we handle negations.</p>
<CENTER>
<TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Axioms of intuitionistic predicate logic">
<CAPTION><B>Axioms of intuitionistic predicate logic</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-4.html">Axiom of Specialization</A></TD>
<TD><FONT COLOR="#006633"><B>ax-4</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD> <A HREF="ax-5.html">Axiom of Quantified Implication</A></TD>
<TD><FONT COLOR="#006633"><B>ax-5</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i5r.html">The converse of ax-5o</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i5r</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>((<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-7.html">Axiom of Quantifier Commutation</A></TD>
<TD><FONT COLOR="#006633"><B>ax-7</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-gen.html">Rule of Generalization</A></TD>
<TD><FONT COLOR="#006633"><B>ax-gen</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
COLOR="#0000FF"><I>φ</I></FONT> => <SPAN >
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ial.html"><SPAN ><I><FONT
COLOR="#FF0000">x</FONT></I></SPAN> is bound in <SPAN ><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></A></TD>
<TD><FONT COLOR="#006633"><B>ax-ial</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ie1.html"><SPAN ><I><FONT
COLOR="#FF0000">x</FONT></I></SPAN> is bound in <SPAN ><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></A></TD>
<TD><FONT COLOR="#006633"><B>ax-ie1</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ie2.html">Define existential
quantification</A></TD>
<TD><FONT COLOR="#006633"><B>ax-ie2</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) ↔ (<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-8.html">Axiom of Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-8</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">z</FONT></I>
→ <I><FONT COLOR="#FF0000">y</FONT></I> = <I><FONT
COLOR="#FF0000">z</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i9.html">Axiom of Existence</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i9</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I></SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-10.html">Axiom of Quantifier
Substitution</A></TD>
<TD><FONT COLOR="#006633"><B>ax-10</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I>
<I><FONT COLOR="#FF0000">y</FONT></I> = <I><FONT
COLOR="#FF0000">x</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-11.html">Axiom of Variable
Substitution</A></TD>
<TD><FONT COLOR="#006633"><B>ax-11</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">y</FONT></I><FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">x</FONT></I>(<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i12.html">Axiom of Quantifier
Introduction</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i12</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I> <I><FONT
COLOR="#FF0000">z</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> (<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">z</FONT></I> <I><FONT COLOR="#FF0000">z</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif> ∨</FONT> <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)))</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-bndl.html">Axiom of Bundling</A></TD>
<TD><FONT COLOR="#006633"><B>ax-bndl</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I> <I><FONT
COLOR="#FF0000">z</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> (<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">z</FONT></I> <I><FONT COLOR="#FF0000">z</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif> ∨</FONT> <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-13.html">Left Membership Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-13</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">z</FONT></I> → <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-14.html">Right Membership Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-14</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">x</FONT></I> → <I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-17.html">Distinctness</A></TD>
<TD><FONT COLOR="#006633"><B>ax-17</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN>,
where <FONT COLOR="#FF0000">x</FONT> does not occur in <FONT
COLOR="#0000FF"><I>φ</I></FONT></TD></TR>
</TABLE>
</CENTER>
<P><A NAME="staxioms"></A><B><FONT COLOR="#006633">Set theory</FONT></B>
uses the formalism of propositional and predicate calculus to assert
properties of arbitrary mathematical objects called "sets." A set can
be an element of another set, and this relationship is indicated by the
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> symbol.
Starting with the simplest mathematical object, called the empty set,
set theory builds up more and more complex structures whose existence
follows from the axioms, eventually resulting in extremely complicated
sets that we identify with the real numbers and other familiar
mathematical objects. These axioms were developed in response to <A
HREF="ru.html">Russell's Paradox</A>, a discovery that revolutionized
the foundations of mathematics and logic.</P>
<P><A NAME="izfaxioms"></A> In the IZF axioms that follow, <I>all set
variables are assumed to be</I> <A HREF="#distinct">distinct</A>. If
you click on their links you will see the explicit distinct variable
conditions.</P>
<CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Intuitionistic Zermelo-Fraenkel Set Theory (IZF)">
<CAPTION><B>Intuitionistic Zermelo-Fraenkel Set Theory (IZF)</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-ext.html">Axiom of Extensionality</A></TD>
<TD NOWRAP><FONT COLOR="#006633"><B>ax-ext</B></FONT></TD>
<TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">x</FONT></I> ↔ <I><FONT COLOR="#FF0000">z</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">y</FONT></I>) →
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-coll.html">Axiom of Collection</A></TD>
<TD><FONT COLOR="#006633"><B>ax-coll</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑎</SPAN> <FONT FACE=sans-serif>∃</FONT><I><FONT
COLOR="#FF0000">y</FONT></I><FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT FACE=sans-serif>∃</FONT><SPAN CLASS=set
STYLE="color:red">𝑏</SPAN><FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT> <SPAN
CLASS=set STYLE="color:red">𝑎</SPAN> <FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑏</SPAN> <FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-sep.html">Axiom of Separation</A></TD>
<TD><FONT COLOR="#006633"><B>ax-sep</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I> ↔ (<I><FONT COLOR="#FF0000">x</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">z</FONT></I> <FONT
FACE=sans-serif>∧</FONT> <FONT
COLOR="#0000FF"><I>φ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-pow.html">Axiom of Power Sets</A></TD>
<TD><FONT COLOR="#006633"><B>ax-pow</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">w</FONT></I>(<I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I> → <I><FONT COLOR="#FF0000">w</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I>) →
<I><FONT COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-pr.html">Axiom of Pairing</A></TD>
<TD><FONT COLOR="#006633"><B>ax-pr</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">z</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">w</FONT></I>((<I><FONT
COLOR="#FF0000">w</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> <I><FONT COLOR="#FF0000">w</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>) → <I><FONT COLOR="#FF0000">w</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-un.html">Axiom of Union</A></TD>
<TD><FONT COLOR="#006633"><B>ax-un</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">w</FONT></I>(<I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∧</FONT> <I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">x</FONT></I>) → <I><FONT COLOR="#FF0000">z</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-setind.html">Axiom of Set Induction</A></TD>
<TD><FONT COLOR="#006633"><B>ax-setind</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><SPAN CLASS=set
STYLE="color:red">𝑎</SPAN>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑎</SPAN> [<I><FONT COLOR="#FF0000">y</FONT></I> /
<SPAN CLASS=set STYLE="color:red">𝑎</SPAN>]<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>) → <FONT
FACE=sans-serif>∀</FONT><SPAN CLASS=set
STYLE="color:red">𝑎</SPAN><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD>
</TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-iinf.html">Axiom of Infinity</A></TD>
<TD><FONT COLOR="#006633"><B>ax-iinf</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(∅
<FONT FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif>∧</FONT> <FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">y</FONT></I>(<I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I> → suc
<I><FONT COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">x</FONT></I>))</SPAN>
</TD></TR>
</TABLE></CENTER>
<P>We develop set theory based on the Intuitionistic Zermelo-Fraenkel
(IZF) system, mostly following the IZF axioms as laid out in [Crosilla].
The one exception to the statement that we use IZF is that a few sections
develop set theory using Constructive Zermelo-Fraenkel (CZF), also described
in Crosilla. These sections start at ~ wbd (including the section header right
before it) and the biggest complication is the machinery to classify formulas
as bounded formulas, for purposes of the Axiom of Restricted Separation
~ ax-bdsep .</P>
<P></P><HR NOSHADE SIZE=1><A NAME="theorems"></A><B><FONT COLOR="#006633">A
Theorem Sampler</FONT></B>
<P></P><CENTER><FONT COLOR="#006633"><I>From a psychological point of view,
learning constructive mathematics is agonizing, for it requires one to
first unlearn certain deeply ingrained intuitions and
habits acquired during classical mathematical training.</I>
<BR> —Andrej Bauer</FONT></CENTER>
<P>Listed here are some examples of starting points for your journey
through the maze. Some are stated just as they would be in a
non-constructive context; others are here to highlight areas which
look different constructively.
You should study some simple proofs from
propositional calculus until you get the hang of it. Then try some
predicate calculus and finally set theory.</P>
<P>The <A HREF="mmtheorems.html">Theorem List</A> shows the complete set of
theorems in the database. You may also find the <A
HREF="mmbiblio.html">Bibliographic Cross-Reference</A> useful.</P>
<P><TABLE BORDER=0><TR><TD VALIGN=TOP WIDTH="50%"><TR><TD VALIGN=TOP WIDTH="50%">
<B>Propositional calculus</B>
<MENU>
<LI>Law of identity ~ idALT</LI>
<LI>Praeclarum theorema ~ anim12</LI>
<LI>Contraposition introduction ~ con3</LI>
<LI>Double negation introduction ~ notnot</LI>
<LI>Triple negation ~ notnotnot</LI>
<LI>Definition of exclusive or ~ df-xor</LI>
<LI>Negation and the false constant ~ dfnot</LI>
</MENU>
<B>Predicate calculus</B>
<MENU>
<LI>Existential and universal quantifier swap ~ 19.12</LI>
<LI>Existentially quantified implication ~ 19.35-1</LI>
<LI><I>x</I> = <I>x</I> ~ equid</LI>
<LI>Definition of proper substitution ~ df-sb</LI>
<LI>Double existential uniqueness ~ 2eu7</LI>
</MENU>
<B>Set theory</B>
<MENU>
<LI>Commutative law for union ~ uncom</LI>
<LI>A basic relationship between class and wff
variables ~ abeq2</LI>
<LI>Two ways of saying "is a set" ~ isset</LI>
<LI>The ZF axiom of foundation implies excluded middle ~ regexmid</LI>
<LI>Russell's paradox ~ ru</LI>
<LI>Ordinal trichotomy implies excluded middle ~ ordtriexmid</LI>
<LI>Mathematical (finite) induction ~ findes</LI>
<LI>Peano's postulates for arithmetic:
~ peano1 ~ peano2 ~ peano3 ~ peano4 ~ peano5</LI>
<LI>Two natural numbers are either equal or not equal ~ nndceq (a special case of the law of the excluded middle which we can prove).</LI>
<LI>A natural number is either zero or a successor ~ nn0suc</LI>
<LI>The axiom of choice implies excluded middle ~ acexmid</LI>
<LI>Burali-Forti paradox ~ onprc</LI>
<LI>Transfinite induction ~ tfis3</LI>
<LI>Closure law for ordinal addition ~ oacl</LI>
</MENU>
<B>Real and complex numbers</B>
<MENU>
<LI>Archimedean property of real numbers ~ arch</LI>
<LI>Properties of apartness:
~ apirr ~ apsym ~ apcotr ~ apti</LI>
<LI>The square root of 2 is irrational ~ sqrt2irrap (a
different statement than "The square root of 2
is not rational" ~ sqrt2irr )</LI>
<LI>Convergence of a sequence of complex
numbers ~ climcvg1n given a condition on the rate of convergence</LI>
<LI>Triangle inequality for absolute
value ~ abstrii</LI>
<LI>The maximum of two real numbers ~ maxleb</LI>
</MENU>
</TD></TR></TABLE>
<HR NOSHADE SIZE=1><A NAME="intuitionize"></A><B><FONT
COLOR="#006633">How to intuitionize classical proofs</FONT></B>
<P>For the student or master of classical mathematics, constructive mathematics
can be baffling. One can get over some of the intial hurdles of understanding
how constructive mathematics works and why it might be interesting by reading
<A HREF="#Bauer">[Bauer]</A> but that work does little to explain in concrete terms how to write
proofs in intuitionistic logic. Fortunately, metamath helps with one of the
biggest hurdles: noticing when one is even using the law of the excluded
middle or the axiom of choice. But suppose you have a classical proof from
the Metamath Proof Explorer and it fails to verify when you copy it over to
the Intuitionistic Logic Explorer. What then? Here are some rules of thumb
in converting classical proofs to intuitionistic ones.</P>
<UL>
<LI>If you see case elimination ( pm2.61 or its variants) you'll probably end up with two theorems for the two cases. In particular, if the cases were ` A e. _V ` and ` -. A e. _V ` you probably just care about the ` A e. _V ` case.
On the other hand, if the proposition being eliminated is decidable
(for example due to ~ nndceq , ~ zdceq , ~ zdcle , ~ zdclt , ~ eluzdc , or ~ fzdcel ),
then case elimination will work using theorems such as ~ df-dc and ~ mpjaodan .</LI>
<LI>
Non-empty almost always needs to be changed to inhabited (those terms are defined at ~ n0rf ).
</LI>
<LI>If the original proof relied on propositional/predicate logic which isn't a theorem of intuitionistic logic, maybe there is a way of expressing the logic more directly. This is perhaps the hardest one to put in cookbook form: you might need to try some things and see if anything works.</LI>
<LI>If the original proof relied on df-ex so that it could prove a theorem for ` A. ` and then get ` E. ` for free (or vice versa), instead go look at the original proof and try to come up the analogues to each step for the other quantifier (for example, ~ cbvrexcsf , ~ sbcrext , ~ rexxpf ). Similarly, if you have a theorem for ` <_ ` and are trying to prove the corresponding theorem for ` < ` you'll probably need to use analogous steps rather than negation (examples: ~ leaddsub , ~ ltsub1 , ~ ltsub2 ).</LI>
<LI>If you are dealing with a definition, try to find the best constructive definition from the literature ([HoTT] book, Stanford Encyclopedia of Philosophy, [Bauer], etc). Once you pick a definition, that'll affect the proofs which rely on that definition.</LI>
<LI>If there is case elimination on whether variables are distinct, most of the time you just need the variant with distinct variables. Sometimes you can then remove the constraint with a temporary variable (e.g. the various sbco2* variants, ~ nfralya , ~ r19.3rm ).</LI>
<LI>Sometimes one direction of a biconditional holds, or subset holds instead of equality. You might be able to keep the easy direction and worry about the other one later.</LI>
<LI>
If there is case elimination sometimes only one of the two cases is possible.
For example, in ~ mosubopt the rest of the formula being proved constrains
which case matters.
</LI>
<LI>
If you need an additional condition (for example, because the original proof
used case elimination) and you are proving a biconditional, consider whether
both sides of the biconditional imply the condition. If so, you'll be able to
prove the biconditional with that condition as an antecedent, and then use
~ pm5.21nii or one of its variants to remove the antecedent (example:
~ elxp4 ).
</LI>
<LI>
If your proof relies on dveeq2 try ~ dveeq2or and likewise for the other things
downstream of ~ ax-i12 or ~ ax-bndl .
</LI>
<LI>
If you have a disjunction, be reluctant to turn it into an implication using
~ ord and the like. Instead, show that each disjunct implies what you are
trying to prove and use ~ jaoi to join those two implications into something
which can hook up to the disjunction.
</LI>
<LI>
Disjunctive syllogism holds in intuitionistic logic and we state it a few ways
(for example ~ orel1 and ~ ecased ) but we don't have a wide variety of
convenience theorems. Unless we add those, you'll use ~ ord or something
similar followed by ~ mpd or something similar. This may add a few steps but
they are straightforward ones.
</LI>
<LI>
If your proof is doing tricky things perhaps in the interest of shortness, try
just expanding the definitions and applying logic in a straightforward way. See
if this gets you a working (although perhaps longer) proof.
</LI>
<LI>
If your proof relies on a biconditional in set.mm which isn't in iset.mm, see
if one direction is in iset.mm and see which direction your proof is using. For
example ~ 19.35-1 or ~ exnalim .
</LI>
<LI>
If you are doing things with inhabited classes (beyond just applying existing
inhabited class theorems), you may be able to dig up some predicate logic you
haven't used in a while (e.g. ~ raaan ).
</LI>
<LI>
Consider the possibility of giving up. Some things just won't have
intuitionistic proofs. The more it looks like excluded middle or other
non-intuitionistic statements, the more likely you are dealing with one of
these. But it can be hard to have good intuition about this. In some cases it
may be possible to ask "can I use this statement to prove ` ph \/ -. ph ` for an
arbitrary proposition" (see ~ ordtriexmid for example ), but this is not always
an easy technique to apply.
</LI>
<LI>
Switching between ~ 2th and ~ 2false might help (e.g. ~ dfnul2 , ~ dfnul3 ,
~ rab0 ).
</LI>
<LI>
In many cases statements which are equivalent in classical logic become several
non-equivalent statements (e.g. exclusive or, ordinals, non-empty versus
inhabited, apartness versus negated equality). This is usually a good place to
look for a literature reference, but don't be afraid to change the statement
being proved to "what you really meant is X" as appropriate.
</LI>
<LI>
If a statement has multiple equivalences in set.mm (e.g. mof and mo3 , or
dffun2 and dffun3 ) and only some of them in iset.mm, sometimes a pretty
similar proof will work (that is, which one to use in the original proof may
have been a fairly arbitrary choice).
</LI>
<LI>
A number of theorems related to functions (especially ovex and fvex ) in
set.mm perform case elimination based on whether we are evaluating the function
within its domain or outside it. The most straightforward solution is to
use ~ fnovex or ~ funfvex which only work within the domain. Using these
may involve rearranging logic, for example by changing ~ rexlimivw to
~ rexlimdva (example: ~ ovelrn or indeed most uses of ~ fnovex and ~ funfvex ).
If a function value is inhabited, we know we are evaluating it within
its domain by ~ relelfvdm .
</LI>
<LI>
With excluded middle, ` (/) e. A ` and ` A =/= (/) ` are equivalent (where
` A ` is an ordinal). In such theorems, ` (/) e. A ` is generally the more
interesting condition constructively.</LI>
<LI>
Reverse closure in set.mm uses excluded middle (for example ovrcl or ndmfvrcl
). The most general way to handle this is to add more conditions that we are
evaluating operations within
their domains (for example set.mm's addasspi versus iset.mm's ~ addasspig in
which conditions such as ` A e. N. ` are added, or set.mm's ltbtwnnq versus
iset.mm's ~ ltbtwnnqq , in which ` E. x ` is changed to ` E. x e. Q. ` ).
If the result of applying a function is inhabited, then we know we applied
it within its domain - that is ~ relelfvdm or ~ elmpocl may be useful.
(These thoughts apply to operations - in at least one place, ~ dvdszrcl ,
set.mm uses the term "reverse closure" for binary relations - but this is
a lot more like ~ brel than like the reverse closure theorems described
above).
</LI>
<LI>
With excluded middle not equal (` =/= `) and apart (` =//= `) are equivalent.
When working with real and complex numbers, apartness is almost always what
you want. See ~ df-ap for more on apartness.
</LI>
<LI>
Given a theorem of the form ` X = Y <-> Z = W ` we can derive
` X =/= Y <-> Z =/= W ` but in many contexts what we really want is
` X =//= Y <-> Z =//= W ` . See if the version with apartness already
exists and if not consider adding it (building from basic apartness
theorems like ~ apadd1 and ~ apmul1 for example). Once you have proved
the version using apartness you can use it to prove the version with
equality if you don't already have it, using ~ notbid and ~ apti .
</LI>
<LI>Exclusive or ( ` ph \/_ ps ` ) is equivalent to ` ph <-> -. ps `
given excluded middle but we just have one direction ( ~ xorbin ).
Consider intuitionizing ` ph <-> -. ps ` as ` ph \/_ ps `
(example: ~ rpnegap ).</LI>
<LI>The expression ` ( _I `` A ) ` evaluates to ` A ` if ` A ` is
a set (by ~ fvi ) and the empty set if ` A ` is a proper class (by
~ fvprc ). However, combining these two facts and thus using ` _I `
to protect against proper classes does not seem to be possible
without excluded middle (set.mm examples: sumeq2ii and strfvi ).
Usually you will end up adding a ` A e. _V ` condition in such
cases.</LI>
<LI>
If you get stuck, ask! (for example in a GitHub issue or on the mailing list).
We have a number of contributors who have experience in constructive
mathematics in general, or iset.mm in particular, and one of the best things
about doing/learning mathematics in metamath is the collaborative nature of how
we develop it.
</UL>
<HR NOSHADE SIZE=1><A NAME="setmm"></A>
<B><FONT COLOR="#006633">Metamath Proof Explorer cross