[FOM] 598: Finite Axiomatizations

[Harvey Friedman]

It is well known that BZ = bounded Zermelo set theory, and NBG = von
Neumman Bernay's Goedel class theory, are finitely axiomatizable. I
want to give a memorable formulation of this that I haven't seen
mentioned in the literature. Perhaps readers can refer me to a place
where this has been done?

1. BOUNDED Z
2. NBG

1. BOUNDED Z

We use the primitives epsilon, =. Bounded Separation is the scheme

{x in A: phi} exists
(thereexists y)(forall x)(x in y iff x in A and phi)

where y,A are not in phi, and all quantifiers in phi are relativized to A.

DEFINITION 1. n quantifier m parameter bounded separation is bounded
separation for formulas phi with at most n quantifiers and at most m
distinct free variables.

In particular, we do not count the domain A of the bounded separation.
Note that up to logical equivalence, there are only finitely many
axioms in n quantifier m parameter bounded separation. This is because
we can place the phi in prenex form and put the quantifier free part
in disjunctive normal form.

THEOREM 1. Bounded Separation and 4 quantifier 2 parameter Bounded
Separation are equivalent over Extensionality, Pairing, Pairwise
Union, Power Set.

So for all of the usual forms of BZ = Bounded Zermelo set theory, we
can use 4 quantifier 2 parameter Bounded Separation.

2. NBG

We again use primitives epsilon, =. We use the standard abbreviation
M(x) for (therexists y)(x in y).

over Extensionality, Pairing for M's, Union for M's, Power Set for
M's. Comprehension is the scheme

The class of all M's with phi exists
(therexists A)(forall x)(x in A iff M(x) and phi)

where A is not in phi, and all quantifiers in phi are relativized to
the x with M(x).

DEFINITION 2. n quantifier m parameter comprehension is comprehension
for formulas phi with at most n quantifiers and at most m distinct
free variables.

THEOREM 2. Comprehension and 4 quantifier 2 parameter Comprehension
are equivalent over M Extensionality, M pairing, M Pairwise Union, M
Power Set.

M Ext. Any two M's with the same elements are equal.
M Pairing. Any two M's have an unordered pair which is also an M.
M Pairwise Union. Any two M's have a pairwise union which is also an M.
M Power Set. Any two M's have a power set which is also an M.

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My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 598th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM
545: Set Theoretic Consistency/SRM/SRP  9/14/14  10:06PM
546: New Pi01/solving CH  9/26/14  12:05AM
547: Conservative Growth - Triples  9/29/14  11:34PM
548: New Explicitly Pi01  10/4/14  8:45PM
549: Conservative Growth - beyond triples  10/6/14  1:31AM
550: Foundational Methodology 1/Maximality  10/17/14  5:43AM
551: Foundational Methodology 2/Maximality  10/19/14 3:06AM
552: Foundational Methodology 3/Maximality  10/21/14 9:59AM
553: Foundational Methodology 4/Maximality  10/21/14 11:57AM
554: Foundational Methodology 5/Maximality  10/26/14 3:17AM
555: Foundational Methodology 6/Maximality  10/29/14 12:32PM
556: Flat Foundations 1  10/29/14  4:07PM
557: New Pi01  10/30/14  2:05PM
558: New Pi01/more  10/31/14 10:01PM
559: Foundational Methodology 7/Maximality  11/214  10:35PM
560: New Pi01/better  11/314  7:45PM
561: New Pi01/HUGE  11/5/14  3:34PM
562: Perfectly Natural Review #1  11/19/14  7:40PM
563: Perfectly Natural Review #2  11/22/14  4:56PM
564: Perfectly Natural Review #3  11/24/14  1:19AM
565: Perfectly Natural Review #4  12/25/14  6:29PM
566: Bridge/Chess/Ultrafinitism 12/25/14  10:46AM
567: Counting Equivalence Classes  1/2/15  10:38AM
568: Counting Equivalence Classes #2  1/5/15  5:06AM
569: Finite Integer Sums and Incompleteness  1/515  8:04PM
570: Philosophy of Incompleteness 1  1/8/15 2:58AM
571: Philosophy of Incompleteness 2  1/8/15  11:30AM
572: Philosophy of Incompleteness 3  1/12/15  6:29PM
573: Philosophy of Incompleteness 4  1/17/15  1:44PM
574: Characterization Theory 1  1/17/15  1:44AM
575: Finite Games and Incompleteness  1/23/15  10:42AM
576: Game Correction/Simplicity Theory  1/27/15  10:39 AM
577: New Pi01 Incompleteness  3/7/15  2:54PM
578: Provably Falsifiable Propositions  3/7/15  2:54PM
579: Impossible Counting  5/26/15  8:58PM
580: Goedel's Second Revisited  5/29/15  5:52 AM
581: Impossible Counting/more  6/2/15  5:55AM
582: Link+Continuation Theory  1  6/21/15  5:38PM
583: Continuation Theory 2  6/23/15  12:01PM
584: Finite Continuation Theory 3   6/26/15  7:51PM
585: Finite Continuation Theory 4  6/29/15  11:23PM
586: Finite Continuation Theory 5  6/20/15  1:32PM
587: Finite Continuation Theory 6  7/1/15  11:39PM
588: Finite Continuation Theory 7  7/2/15  2:44PM
589: Finite Continuation Theory 8  7/4/15  6:51PM
590: Finite Continuation Theory 9  7/6/15  5:20PM
591: Finite Continuation Theory 10  7/12/15  3:38PM
592: Finite Continuation Theory 11/perfect?  7/29/15  4:30PM
593: Finite Continuation Theory 12/perfect?  8/23/15  9:47PM
594: Finite Continuation Theory 13/perfect?  8/4/15  1:44PM
595: Finite Continuation Theory 14/perfect?  8/5/15  8:23PM
596: Finite Continuation Theory 15/perfect?  8/8/15 12:35AM
597: Finite Continuation Theory 16/perfect?

Harvey Friedman