The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: An equivalent of MK-Foundation-Choice
Replies: 0  

Advanced Search

Back to Topic List Back to Topic List

Posts: 2,665
Registered: 6/29/07
An equivalent of MK-Foundation-Choice
Posted: Feb 20, 2013 5:58 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

This is just a cute result.

The following theory is equal to MK-Foundation-Choice

Language: FOL(=,e)

Define: Set(x) iff Ey. x e y

Axioms: ID axioms+

1.Extensionality: (Az. z e x <-> z e y) -> x=y

2. Construction: if phi is a formula in which x is not free,
then (ExAy.y e x<->Set(y)&phi) is an axiom

3. Pairing: (Ay. y e x -> y=a or y=b) -> Set(x)

4. Set(x) <-> Ey. y is set sized & Azex(Emey(z<<m))

where y is set sized iff Es. Set(s) & y =< s
and z<<m iff z =<m & AneTC(z).n =<m

TC(z) stands for 'transitive closure of z' defined in the usual manner
as the minimal transitive class having z as subclass of; transitive of
course defined as a class having all its members as subsets of it.

y =< s iff Exist f. f:y-->s & f is injective.


Of course this theory PROVES the consistency of ZFC.
Proofs had all been worked up in detail. It is an enjoying experience
to try figure them out.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.