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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"https://www.w3.org/TR/html4/loose.dtd">
<!-- The file mmnf.html is generated from mmnf.raw.html -
see the regen-from-raw script for details -->
<HTML LANG="EN-US">
<HEAD>
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<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 Page - New Foundations Explorer</TITLE>
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<TD ALIGN=LEFT VALIGN=TOP><A HREF="../index.html"><IMG SRC="nf.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>New Foundations 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="elopprim.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> >
NFE Home >
<A HREF="mmtheorems.html">Th. List</A>
</FONT>
</TD>
</TR>
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<HR NOSHADE SIZE=1>
<CENTER>
<B><FONT COLOR="#006633">Created by Scott Fenton</FONT></B>
</CENTER>
<HR NOSHADE SIZE=1>
<B><FONT COLOR="#006633">New Foundations Proof Explorer</FONT></B>
<TABLE>
<TR><TD ROWSPAN=2>
<I>New Foundations</I>
(<A HREF="http://en.wikipedia.org/wiki/New_Foundations">Wikipedia</A>
[external], <A HREF="http://plato.stanford.edu/entries/quine-nf/">
Stanford Encyclopedia of Philosophy</A> [external]) is an alternative set
theory to the Zermelo-Fraenkel set theory presented in the regular Metamath
Proof Explorer.
Unlike the Zermelo-Fraenkel system with the Axiom of Choice
(known as ZFC), New Foundations is a direct derivative of
the set theory originally presented in <I>Principia Mathematica</I>.
</TD></TR></TABLE>
<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="#strat">Stratification</A></LI>
<LI> <A HREF="#axioms">The axioms</A></LI>
<LI> <A HREF="#theorems">Some theorems</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="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/nf.mm">Metamath database nf.mm (ASCII file)</A></LI>
</MENU>
<B><FONT COLOR="#006633">External links</FONT></B>
<MENU>
<LI> <A HREF="https://github.com/sctfn/metamath-nf/">GitHub repository</A></LI>
</MENU>
</TD>
</TR></TABLE>
<HR NOSHADE SIZE=1><A NAME="strat"></A><B><FONT COLOR="#006633">
Stratification</FONT></B>
<p>
In <I>Principia Mathematica</I>, Russell and Whitehead used a typing system
to avoid the paradoxes of naive set theory,
rather than restrict the size of sets (as Zermelo-Fraenkel theory does).
This typing system was
eventually refined by Russell down to Typed Set Theory (TST).
In TST, unlimited comprehension is allowed
(approximately,
A ` e. ` ` _V ` is a theorem).
TST avoids the standard paradoxes by being a multi-sorted system.
That is, there are variables of type 0, 1, 2,... The WFFs are
restricted so that you must say
x[n] = y[n] and x[n] ` e. ` y[n+1], where n
is a variable type.
This means, among other things, that
x ` e. ` x is not a well-formed formula,
so we can't even sensibly speak of the Russell class.
Thus TST counters <A HREF="ru.html">Russell's Paradox</A>.
<p>
Now, consider introducing virtual classes into this theory. You need
to say things like V[n] = { x[n] | x[n] = x[n] } for each type n.
This leads to a "hall of mirrors" type situation: each named object is
duplicated for each type.
<p>
Quine noticed this and proposed collapsing
the whole theory into a one-sorted set theory, where the comprehension
axiom is restricted to formulas where you could theoretically introduce
subscripts to make the formula a WFF of TST.
Quine described this approach in a paper titled
"New Foundations for Mathematical Logic,"
so this approach is now called "New Foundations" (NF)
<A HREF="#Quine2">[Quine2]</A>.
For more details, see the
<A HREF="http://en.wikipedia.org/wiki/New_Foundations">Wikipedia
article on NF</A>.
<HR NOSHADE SIZE=1><A NAME="axioms"></A><B><FONT COLOR="#006633">
The axioms</FONT></B>
<p>
The axioms begin with traditional axioms for
classical first order logic with equality.
See the regular
<a href="/mpeuni/mmset.html">Metamath Proof Explorer</a>
for discussions about these
axioms and some of their implications.
<p>
The key axioms specific to NF are
<a href="ax-ext.html">extensionality</a>
(two sets are identical if they contain the same elements) and
a comprehension schema.
Extensionality is formally defined as:
<CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Extensionality">
<CAPTION><B>Extensionality</B></CAPTION>
<TR><TH>Name</TH><TH>Ref</TH><TH>Expression</TH></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-ext.html">Axiom of Extensionality</A></TD>
<TD><FONT COLOR="#006633"> ~ ax-ext </FONT></TD>
<TD>` |- ( A. z ( z e. x <-> z e. y ) -> x = y ) ` </TD></TR>
</TABLE>
</CENTER>
<p>
The comprehension schema is stated using the concept of stratified formula;
the approach is the Stratification Axiom from [Quine2].
In short,
a well-formed formula using only propositional symbols, predicate
symbols, and ` e. ` is "stratified" iff you can make a (metalogical)
mapping from the variables to the natural numbers such that any formulas
of the form x = y have the same number,
and any formulas of the form
x ` e. ` y have
x as one less than y.
Quine's stratification axiom states that there is a
set corresponding to any stratified formula.
We use Hailperin's axioms and prove existence of stratified sets using
Hailperin's algorithm.
Thus the stratification axiom of [Quine2] is
implemented in this formalization using the axioms P1 through P9 from [Hailperin] and
the <a href="ax-sn.html">Axiom of Singleton ax-sn</a>:
<CENTER>
<TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="List of New Foundations Stratification Axioms">
<CAPTION><B>New Foundations Stratification Axioms</B></CAPTION><TR
ALIGN=LEFT><TD><B>Name</B></TD><TD><B>
Ref</B></TD><TD><B>Expression (see link for any distinct variable requirements)
</B></TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-nin.html">Axiom of Anti-Intersection (P1)</A></TD>
<TD> ~ ax-nin </TD>
<TD>` |- E. z A. w ( w e. z <-> ( w e. x -/\ w e. y ) ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-si.html">Axiom of Singleton Image (P2)</A></TD>
<TD> ~ ax-si </TD>
<TD>` |- E. y A. z A. w ( << { z } , { w } >> e. y <-> << z , w >> e. x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-sn.html">Axiom of Singleton (not directly stated in Hailperin)</A></TD>
<TD> ~ ax-sn </TD>
<TD>` |- E. y A. z ( z e. y <-> z = x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-ins2.html">Axiom of Insertion Two (P3)</A></TD>
<TD> ~ ax-ins2 </TD>
<TD>` |- E. y A. z A. w A. t ( << { { z } } , << w , t >> >> e. y <-> << z , t >> `
` e. x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-ins3.html">Axiom of Insertion Three (P4)</A></TD>
<TD> ~ ax-ins3 </TD>
<TD>` |- E. y A. z A. w A. t ( << { { z } } , << w , t >> >> e. y <-> << z , w >> `
` e. x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-xp.html">Axiom of Cross Product (P5)</A></TD>
<TD> ~ ax-xp </TD><TD>` |- E. y A. z ( z e. y <-> E. w E. t ( z = << w , t >> /\ t e. x ) ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-typlower.html">Axiom of Type Lowering (P6)</A></TD>
<TD> ~ ax-typlower </TD>
<TD>` |- E. y A. z ( z e. y <-> A. w << w , { z } >> e. x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-cnv.html">Axiom of Converse (P7)</A></TD>
<TD> ~ ax-cnv </TD>
<TD>` |- E. y A. z A. w ( << z , w >> e. y <-> << w , z >> e. x ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-1c.html">Axiom of Cardinal One (P8)</A></TD>
<TD> ~ ax-1c </TD>
<TD>` |- E. x A. y ( y e. x <-> E. z A. w ( w e. y <-> w = z ) ) ` </TD></TR>
<TR BGCOLOR="#EEFFFA" ALIGN=LEFT>
<TD><A HREF="ax-sset.html">Axiom of Subset Relationship (P9)</A></TD>
<TD> ~ ax-sset </TD>
<TD>` |- E. x A. y A. z ( << y , z >> e. x <-> A. w ( w e. y -> w e. z ) ) ` </TD></TR>
</TABLE>
</CENTER>
<p>
The usual definition of the ordered pair, first proposed by Kuratowski
in 1921 and used in the regular Metamath Proof Explorer,
has a serious drawback for NF and related theories that use stratification.
The Kuratowski ordered pair
is defined as << x , y >> = { { x } , { x , y } }.
This leads to the ordered pair having a type two greater than its arguments.
For example, z in << << x , y >> , z >>
would have a different type than x and y,
which makes multi-argument functions extremely awkward to work with.
Operations such as "1st" and complex "+" would not form sets
in NF with the Kuratowski ordered pairs.
<p>
In contrast, the Quine definition of ordered pairs,
defined in definition <a href="df-op.html">df-op</a>, is type level.
That is, <. x , y >. has the same type as x and y,
which means that the same holds of <. <. x , y >. , z >.
This means that "1st" is a set with the Quine definition,
as is complex "+".
The Kuratowski ordered pair is defined
(as <a href="df-opk.html">df-opk</a>), because
it is a simple definition that can be used by the set construction axioms
that follow it, but for typical uses the Quine definition of ordered pairs
<a href="df-op.html">df-op</a> is used instead.
<p>
Perhaps one of the most remarkable aspects of NF is that the
<A HREF="nchoice.html">Axiom of Choice (an axiom of the widely-used ZFC system)
can be disproven in NF</A>, a result
proven in [Specker].
As a corollary, NF proves infinity.
<p>
There are several systems <i>related</i> to NF.
In particular, NFU is a small modification of NF that also allows
urelements (multiple distinct objects lacking members).
NFU corresponds to a modified type theory TSTU,
where type 0 has urelements, not just a single empty set.
NFU is consistent with both Infinity and Choice, so both can be added to NFU.
NFU + Infinity + Choice has the same consistency strength
as the theory of types with the Axiom of Infinity.
NFU + Infinity + Choice has been extended further, e.g., with various strong
axioms of infinity (similar to ways that ZFC has been extended).
Randall Holmes states that <a href="http://math.boisestate.edu/~~holmes/holmes/nf.html#Consistent">"Extensions of NFU
are adequate vehicles for mathematics in a basically familiar style"</a>.
NFU is not further discussed here.
<p>
A fair amount of the definitions and
theorems (notably the ones on boolean set operations)
are taken verbatim from the regular Metamath Proof Explorer
source file set.mm (based on ZFC).
This also follows the development in [Rosser] fairly closely.
An unusual aspect is the
<a href="df-tcfn.html">stratified T-raising function TcFn</a>.
The work to specifically formalize New Foundations in metamath
was originally created by Scott Fenton.
Those who are interested in New Foundations may want to look at the
<a href="http://math.boisestate.edu/~~holmes/holmes/nf.html">New Foundations
home page</a>, as well as a
<a href="http://math.boisestate.edu/~~holmes/">proof of the consistency of
New Foundations by Randall Holmes</a>.
The descriptions given here are based on a
<a href="https://groups.google.com/forum/#!topic/metamath/yj1j4Ebb1bI">discussion on the metamath mailing list</a>.
<HR NOSHADE SIZE=1><A NAME="theorems"></A><B><FONT COLOR="#006633">
Some theorems</FONT></B>
<MENU>
<LI> <A HREF="1p1e2c.html">Proof of <I>Principia
Mathematica</I>'s version of 1+1=2</A></LI>
<LI> <A HREF="vinf.html">Proof of the Axiom of Infinity</A></LI>
<LI> <A HREF="ru.html">Russell's Paradox</A></LI>
<LI> <A HREF="vvex.html">Proof that the universal class exists</A></LI>
<LI> <A HREF="nchoice.html">Disproof of the Axiom of Choice</A></LI>
</MENU>
<HR NOSHADE SIZE=1><A NAME="bib"></A><B><FONT
COLOR="#006633">Bibliography</FONT></B>
<OL>
<LI><A NAME="BellMachover"></A> [BellMachover] Bell, J. L., and M.
Machover, <I>A Course in Mathematical Logic,</I> North-Holland,
Amsterdam (1977) [QA9.B3953].</LI>
<LI><A NAME="ChoquetDD"></A> [ChoquetDD] Choquet-Bruhat, Yvonne and Cecile
DeWitt-Morette, with Margaret Dillard-Bleick, <I>Analysis, Manifolds and
Physics,</I> Elsevier Science B.V., Amsterdam (1982) [QC20.7.A5C48
1981].</LI>
<LI><A NAME="Eisenberg"></A> [Eisenberg] Eisenberg, Murray, <I>Axiomatic Theory of
Sets and Classes,</I> Holt, Rinehart and Winston, Inc., New York (1971)
[QA248.E36].</LI>
<LI><A NAME="Enderton"></A> [Enderton] Enderton, Herbert B., <I>Elements of Set
Theory,</I> Academic Press, Inc., San Diego, California (1977)
[QA248.E5].</LI>
<LI><A NAME="Gleason"></A> [Gleason] Gleason, Andrew M., <I>Fundamentals of
Abstract Analysis,</I> Jones and Bartlett Publishers, Boston (1991)
[QA300.G554].</LI>
<LI><A NAME="Hailperin"></A> [Hailperin] Hailperin, Theodore, "A Set of
Axioms for Logic," <I>Journal of Symbolic Logic,</I> 9:1-14 (1944) [BC1.J6].</LI>
<LI><A NAME="Hamilton"></A> [Hamilton] Hamilton, A. G., <I>Logic for
Mathematicians,</I> Cambridge University Press, Cambridge, revised
edition (1988) [QA9.H298 1988].</LI>
<LI><A NAME="Hitchcock"></A> [Hitchcock] Hitchcock, David, <I>The
peculiarities of Stoic propositional logic</I>, McMaster University;
available at <A
HREF="http://www.humanities.mcmaster.ca/~~hitchckd/peculiarities.pdf">
http://www.humanities.mcmaster.ca/~~hitchckd/peculiarities.pdf</A>
(retrieved 3 Jul 2016).</LI>
<LI><A NAME="Holmes"></A> [Holmes] Holmes, Robert, <I>Elementary Set Theory With a Universal Set,</I> Web. Accessed 23 Feb 2015. <A HREF="http://math.boisestate.edu/~~holmes/holmes/head.pdf">Link</A></LI>
<LI><A NAME="Jech"></A> [Jech] Jech, Thomas, <I>Set Theory,</I>
Academic Press, San Diego (1978) [QA248.J42].</LI>
<LI><A NAME="KalishMontague"></A> [KalishMontague] Kalish, D. and R.
Montague, "On Tarski's formalization of predicate logic with
identity," <I>Archiv für Mathematische Logik und
Grundlagenforschung,</I> 7:81-101 (1965) [QA.A673].</LI>
<LI><A NAME="Kunen"></A> [Kunen] Kunen, Kenneth, <I>Set Theory: An
Introduction to Independence Proofs,</I> Elsevier Science B.V.,
Amsterdam (1980) [QA248.K75].</LI>
<LI><A NAME="KuratowskiMostowski"></A> [KuratowskiMostowski] Kuratowski, K.
and A. Mostowski, <I>Set Theory: with an Introduction to
Descriptive Set Theory,</I> 2nd ed., North-Holland,
Amsterdam (1976) [QA248.K7683 1976].</LI>
<LI><A NAME="Levy"></A> [Levy] Levy, Azriel, <I>Basic Set Theory</I>,
Dover Publications, Mineola, N.Y. (2002) [QA248.L398 2002]. </LI>
<LI><A NAME="Lopez-Astorga"></A> [Lopez-Astorga] Lopez-Astorga, Miguel,
"The First Rule of Stoic Logic and its Relationship with the
Indemonstrables", <I>Revista de Filosofía Tópicos</I> (2016);
available at <A HREF="http://www.scielo.org.mx/pdf/trf/n50/n50a1.pdf">
http://www.scielo.org.mx/pdf/trf/n50/n50a1.pdf</A> (retrieved 3 Jul
2016).</LI>
<LI><A NAME="Margaris"></A> [Margaris] Margaris, Angelo, <I>First Order
Mathematical Logic,</I> Blaisdell Publishing Company, Waltham,
Massachusetts (1967) [QA9.M327].</LI>
<LI><A NAME="Megill"></A><A NAME="bibmegill"></A> [Megill] Megill, N.,
"A Finitely Axiomatized Formalization of Predicate Calculus with
Equality," <I>Notre Dame Journal of Formal Logic,</I> 36:435-453
(1995) [QA.N914]; available at <A
HREF="http://projecteuclid.org/euclid.ndjfl/1040149359"
>http://projecteuclid.org/euclid.ndjfl/1040149359</A> (accessed
11 Nov 2014); the <A HREF="../downloads/finiteaxiom.pdf">PDF
preprint</A> has the same content (with corrections) but pages are
numbered 1-22, and the database references use the numbers printed on the
page itself, not the PDF page numbers.</LI>
<LI><A NAME="Mendelson"></A> [Mendelson] Mendelson, Elliott, <I>Introduction to
Mathematical Logic,</I> 2nd ed., D. Van Nostrand (1979) [QA9.M537].</LI>
<LI><A NAME="Monk1"></A> [Monk1] Monk, J. Donald, <I>Introduction to Set
Theory,</I> McGraw-Hill, Inc. (1969) [QA248.M745].</LI>
<LI><A NAME="Monk2"></A> [Monk2] Monk, J. Donald, "Substitutionless
Predicate Logic with Identity," <I>Archiv für Mathematische Logik
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</OL>
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