[FOM] finite axiomatisation

[Ali Enayat]

This is a reply to the following question of Thomas Forster (Aug 18, 2008):


> Can anyone on this list state - and cite! - a theorem to the effect that
> to any recursively axiomatisable theory $T$ in a language $L$ there is a
> finitely axiomatisable theory $T'$ in a suitable language $L'$ with $T'$
> equivalent to $T'$ in some very strong sense.   Somebody must have proved
> a rigorous version of this, and I am hoping that listmembers will know
> who and how.

The theorem you have in mind is due to Kleene, which shows that if T
is formulated in a *finite langauge* L, and is recursively
axiomatized, then there is a finitely axiomatized theorey T', in an
expanded (but still finite) language, such that T' is a *conservative*
extension of T, i.e., T is precisely the set of logical consequences
of T' in the original language L.

Kleene's result was finite tuned by Craig and Vaught; I suggest the
following excellent review of Makkai for more history.


Mihaly Makkai, Reviewed work(s):
Finite Axiomatizability of Theories in the Predicate Calculus Using
Additional Predicate Symbols. by S. C. Kleene,
Finite Axiomatizability Using Additional Predicates. by W. Craig; R. L. Vaught
# The Journal of Symbolic Logic, Vol. 36, No. 2 (Jun., 1971), pp. 334-335.


Best regards,

Ali Enayat


On Mon, Aug 18, 2008 at 12:00 PM, <fom-request at cs.nyu.edu> wrote:
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>   1. finite axiomatisation (Thomas Forster)
>   2.  Uniqueness of hyperreals and the Continuum Hypothesis
>      (Robert L Knighten)
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> ----------------------------------------------------------------------
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> Message: 1
> Date: Mon, 18 Aug 2008 03:29:03 +0100 (BST)
> From: Thomas Forster <T.Forster at dpmms.cam.ac.uk>
> Subject: [FOM] finite axiomatisation
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Message-ID:
>        <Pine.LNX.4.58.0808180325540.10295 at waxwing.dpmms.cam.ac.uk>
> Content-Type: TEXT/PLAIN; charset=US-ASCII
>
>
> Can anyone on this list state - and cite! - a theorem to the effect that
> to any recursively axiomatisable theory $T$ in a language $L$ there is a
> finitely axiomatisable theory $T'$ in a suitable language $L'$ with $T'$
> equivalent to $T'$ in some very strong sense.   Somebody must have proved
> a rigorous version of this, and I am hoping that listmembers will know
> who and how.
>
>    Thomas Forster
>
>
>
>
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> ------------------------------
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> Message: 2
> Date: Sun, 17 Aug 2008 19:32:35 -0700
> From: Robert L Knighten <RLK at knighten.org>
> Subject: [FOM]  Uniqueness of hyperreals and the Continuum Hypothesis
> To: fom at cs.nyu.edu
> Message-ID: <86k5efhx30.fsf at zeus.knighten.org>
>
> When constructing "the" hyperreals as a non-principal ultrapower of the real
> numbers it is frequently mentioned that, assuming the Continuum Hypothesis,
> all such ordered fields are order isomorphic.  It is also occasionally
> mentioned that the converse is true as well -- failure of CH means it is
> possible to construct non-isomorphic hyperreal fields.  I have located a proof
> of the first of these results, but not of the second.  So a question for the
> experts on FOM: is it true, and if so where is a proof to be found.
>
> -- Bob
>
> --
> Robert L. Knighten
> RLK at knighten.org
>
>
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> End of FOM Digest, Vol 68, Issue 11
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