# [FOM] 324: Existential Comprehension

Harvey Friedman friedman at math.ohio-state.edu
Thu Apr 10 22:16:48 EDT 2008

```EXISTENTIAL COMPREHENSION
by
Harvey M. Friedman
April 10, 2008

TOPIC: HOW CAN INFINITE SET THEORY BE VIEWED AS AN EXTRAPOLATION OF
FINITE SET THEORY THROUGH THE ADDITION OF THE AXIOM OF INFINITY?

Some time ago on the FOM email list, I suggested that this question

Here we present an affirmative answer through what we call existential
comprehension.

We take Zermelo set theory to be: extensionality, pairing, union,
separation, and the modern axiom of infinity (using x union {x}). The
primitives are epsilon and =.

Sigma_1 separation is separation for Sigma_1 formulas; i.e., formulas
in epsilon,= with existential quantifiers followed by a formula with
only bounded quantifiers.

No epsilon cycles is the scheme

not (x1 epsilon x2 epsilon ... epsilon xk epsilon x1)

where k >= 1.

V(omega) is the set of all hereditarily finite sets.

1. ECFIN(epsilon,inclusion,=).

Let ECFIN(epsilon,inclusion,=) be the set of all existential
comprehension axioms in epsilon, inclusion, =, that hold in
(V(omega),epsilon).

I.e., we are looking at axioms

{x: (therexists y1,...,yk)(phi)} exists

where phi is quantifier free in epsilon, inclusion, =, that hold in
(V(omega),epsilon). We treat inclusion here as an abbreviation, and
not as a primitive.

THEOREM 1.1. ECFIN(epsilon, inclusion,=) is logically equivalent to
extensionality, pairing, union, power set, Sigma_1 separation, no
epsilon cycles.

COROLLARY 1.2. ECFIN(epsilon, inclusion,=) + Infinity is equivalent to
the following variant of Zermelo set theory: extensionality, pairing,
union, power set, infinity, Sigma_1 separation, no epsilon cycles.

THEOREM 1.3. ECFIN(epsilon, inclusion,=) is decidable, and quite
robust. E.g., we can replace "hold" by provable in various finite set
theories, and also has combinatorial characterizations.

2. ECFIN(epsilon,inclusion,=,R).

Let ECFIN(epsilon,inclusion,=,R) be the set of all existential
comprehension axioms in epsilon, inclusion, =, and the unary schematic
relation symbol R, that hold in (V(omega),epsilon).

I.e., we are looking at axiom schemes

{x: (therexists y1,...,yk)(phi)} exists

where phi is quantifier free in epsilon, inclusion, =, R, that hold in
(V(omega),epsilon). We treat inclusion as an abbreviation, and allow R
to range over all unary relations on V(omega).

For the purpose of logical equivalence below, we treat
ECFIN(epsilon,inclusion,=,R) as a set of axiom schemes which require R
be appropriately subsituted by first order formulas in epsilon,= (or,
equivalently, in epsilon,inclusion,=).

THEOREM 2.1. ECFIN(epsilon,inclusion,=,R) is logically equivalent to
extensionality, pairing, union, power set, separation, no epsilon
cycles.

COROLLARY 2.2. ECFIN(epsilon,inclusion,=,R) + Infinity is logically
equivalent to the following variant of Zermelo set theory:
extensionality, pairing, union, power set, infinity, separation, no
epsilon cycles.

THEOREM 2.3. ECFIN(epsilon,inclusion,=,R) remains logically equivalent
if we require that we use at most 1 existential quantifier in the
formula phi.

THEOREM 2.4. ECFIN(epsilon,inclusion,=,R) is decidable, and quite
robust. E.g., we can replace all unary R by very simple R, or all
"true" by provable in various finite set theories. There are also
combinatorial characterizations.

3. ECFIN(epsilon,inclusion,=,R*).

Let ECFIN(epsilon,inclusion,=,R*) be the set of all existential
comprehension axioms in epsilon, inclusion, =, and the binary
schematic relation symbol R, with functional hypothesis, that hold in
(V(omega),epsilon).
I.e., we are looking at axiom schemes

(forall x)(therexists unique y)(R(x,y)) implies {x: (therexists
y1,...,yk)(phi)} exists

where phi is quantifier free in epsilon, inclusion, =, R*, that hold
in (V(omega),epsilon). We treat inclusion as an abbreviation, and
allow R to range over all binary relations on V(omega).

For the purpose of logical equivalence, we treat
ECFIN(epsilon,inclusion,=,R) as a set of axiom schemes which require R
be appropriately subsituted by first order formulas in epsilon,= (or,
equivalently, in epsilon,inclusion,=).

THEOREM 3.1. ECFIN(epsilon,inclusion,=,R*) is logically equivalent to
extensionality, pairing, union, power set, no epsilon cycles,
separation, replacement.

THEOREM 3.2. ECFIN(epsilon,inclusion,=,R*) + Infinity is equivalent to
ZF with no epsilon cycles replacing foundation.

THEOREM 3.3. ECFIN(epsilon,inclusion,=,R*) remains logically
equivalent if we require that we use at most 1 existential quantifier
in the formula phi.

THEOREM 3.4. ECFIN(epsilon,inclusion,=,R) is decidable, and quite
robust. E.g., we can replace all binary R by very simple R, or all
"true" by provable in various finite set theories. There are also
combinatorial characterizations.

4. ECFIN(epsilon,inclusion,=,rel), ECFIN(epsilon,inclusion,=,rel*).

ECFIN(epsilon,inclusion,=,rel) extends ECFIN(epsilon,inclusion,=,R) to
incorporate infinitely many relations of every arity.

ECFIN(epsilon,inclusion,=,rel*) extends ECFIN(epsilon,inclusion,=,R*)
to incorporate infinitely many relations of every arity with
functional hypothesis.

In the first case, we get logical equivalence with
ECFIN(epsilon,inclusion,=,R). In the second case, we get logical
equivalence with ECFIN(epsilon,inclusion,=,R*).

**********************************

manuscripts. This is the 323rd 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 #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected
from the original.

250. Extreme Cardinals/Pi01  7/31/05  8:34PM
251. Embedding Axioms  8/1/05  10:40AM
252. Pi01 Revisited  10/25/05  10:35PM
253. Pi01 Progress  10/26/05  6:32AM
254. Pi01 Progress/more  11/10/05  4:37AM
255. Controlling Pi01  11/12  5:10PM
256. NAME:finite inclusion theory  11/21/05  2:34AM
257. FIT/more  11/22/05  5:34AM
258. Pi01/Simplification/Restatement  11/27/05  2:12AM
259. Pi01 pointer  11/30/05  10:36AM
260. Pi01/simplification  12/3/05  3:11PM
261. Pi01/nicer  12/5/05  2:26AM
262. Correction/Restatement  12/9/05  10:13AM
263. Pi01/digraphs 1  1/13/06  1:11AM
264. Pi01/digraphs 2  1/27/06  11:34AM
265. Pi01/digraphs 2/more  1/28/06  2:46PM
266. Pi01/digraphs/unifying 2/4/06  5:27AM
267. Pi01/digraphs/progress  2/8/06  2:44AM
268. Finite to Infinite 1  2/22/06  9:01AM
269. Pi01,Pi00/digraphs  2/25/06  3:09AM
270. Finite to Infinite/Restatement  2/25/06  8:25PM
271. Clarification of Smith Article  3/22/06  5:58PM
272. Sigma01/optimal  3/24/06  1:45PM
273: Sigma01/optimal/size  3/28/06  12:57PM
274: Subcubic Graph Numbers  4/1/06  11:23AM
275: Kruskal Theorem/Impredicativity  4/2/06  12:16PM
276: Higman/Kruskal/impredicativity  4/4/06  6:31AM
277: Strict Predicativity  4/5/06  1:58PM
278: Ultra/Strict/Predicativity/Higman  4/8/06  1:33AM
279: Subcubic graph numbers/restated  4/8/06  3:14AN
280: Generating large caridnals/self embedding axioms  5/2/06  4:55AM
281: Linear Self Embedding Axioms  5/5/06  2:32AM
282: Adventures in Pi01 Independence  5/7/06
283: A theory of indiscernibles  5/7/06  6:42PM
284: Godel's Second  5/9/06  10:02AM
285: Godel's Second/more  5/10/06  5:55PM
286: Godel's Second/still more  5/11/06  2:05PM
287: More Pi01 adventures  5/18/06  9:19AM
288: Discrete ordered rings and large cardinals  6/1/06  11:28AM
289: Integer Thresholds in FFF  6/6/06  10:23PM
290: Independently Free Minds/Collectively Random Agents  6/12/06
11:01AM
291: Independently Free Minds/Collectively Random Agents (more)  6/13/06
5:01PM
292: Concept Calculus 1  6/17/06  5:26PM
293: Concept Calculus 2  6/20/06  6:27PM
294: Concept Calculus 3  6/25/06  5:15PM
295: Concept Calculus 4  7/3/06  2:34AM
296: Order Calculus  7/7/06  12:13PM
297: Order Calculus/restatement  7/11/06  12:16PM
298: Concept Calculus 5  7/14/06  5:40AM
299: Order Calculus/simplification  7/23/06  7:38PM
300: Exotic Prefix Theory   9/14/06   7:11AM
301: Exotic Prefix Theory (correction)  9/14/06  6:09PM
302: PA Completeness  10/29/06  2:38AM
303: PA Completeness (restatement)  10/30/06  11:53AM
304: PA Completeness/strategy 11/4/06  10:57AM
305: Proofs of Godel's Second  12/21/06  11:31AM
306: Godel's Second/more  12/23/06  7:39PM
307: Formalized Consistency Problem Solved  1/14/07  6:24PM
308: Large Large Cardinals  7/05/07  5:01AM
309: Thematic PA Incompleteness  10/22/07  10:56AM
310: Thematic PA Incompleteness 2  11/6/07  5:31AM
311: Thematic PA Incompleteness 3  11/8/07  8:35AM
312: Pi01 Incompleteness  11/13/07  3:11PM
313: Pi01 Incompleteness  12/19/07  8:00AM
314: Pi01 Incompleteness/Digraphs  12/22/07  4:12AM
315: Pi01 Incompleteness/Digraphs/#2  1/16/08  7:32AM
316: Shift Theorems  1/24/08  12:36PM
317: Polynomials and PA  1/29/08  10:29PM
318: Polynomials and PA #2  2/4/08  12:07AM
319: Pi01 Incompleteness/Digraphs/#3  2/12/08  9:21PM
320: Pi01 Incompleteness/#4  2/13/08  5:32PM
321: Pi01 Incompleteness/forward imaging  2/19/08  5:09PM
322: Pi01 Incompleteness/forward imaging 2  3/10/08  11:09PM
323: Pi01 Incompleteness/point deletion  3/17/08  2:18PM

Harvey Friedman
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