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Results tagged with lo.logic
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user 9269
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
16
votes
Can there exist a definable "ultrafilter" on the ordinals?
As pointed out by Gabe Goldberg (in the comments) such a $U$ cannot be definable, and as shown by Joel Hamkins' answer, $U$ cannot even have a definable base.
However, if instead of insisting that the …
- 17.7k
8
votes
Is every recursively axiomatizable and consistent theory interpretable in the true arithmeti...
This answer complements the one given by Noah Schweber.
The arithmetized completeness theorem was first proved in the seminal logic text Grundlagen der Mathematik (1928) written by Hilbert and Bernays …
- 17.7k
8
votes
Accepted
Is Presburger arithmetic in stronger logics still complete?
The following theorem of Schmerl and Simpson gives a positive answer to the question; i.e., it shows that $\mathbb{Pres}(\mathcal{R})$ is complete as an $\mathcal{R}$-theory.
In what follows $\maths …
- 17.7k
10
votes
Paris-Harrington via overspill?
This edit includes a fleshed-out version of Smorynski's remark about taking advantage of a nonstandard model of arithmetic (instead of König's lemma) to prove the Paris-Harrington (PH) principle from …
- 17.7k
5
votes
Can uncountable sets be proved to exist in this variant of ZFC with definability restrictions?
The proof below is due to George Boolos, and appears in his article Constructing Cantorian Counterexamples, Journal of Philosophical Logic, Vol. 26, No. 3 (Jun., 1997), pp. 237-23.
Boolos gives an …
- 17.7k
2
votes
Accepted
A conservative extension of Peano Arithmetic
In the classical setting, the result of the type that is asked in the question was established by Harvey Friedman, answering a question of Abraham Robinson; Friedman's results is reported in the follo …
- 17.7k
9
votes
Is this theory synonymous with PA?
Emil Jeřábek has already provided a detailed proof to show that the proposed theory $T$ is not bi-interpretable with $\mathsf{PA}$ (even though it is mutually interpretable with $\mathsf{PA}$).
The po …
- 17.7k
4
votes
Kleene normal form theorem for r.e. relations proven in arithmetical theories
The usual formal systems in which one can carry out basic recursion-theoretic constructions and theorems such as the ones you are asking about are $\mathrm{I}\Sigma_1$ (a fragment of $\mathrm{PA}$ in …
- 17.7k
4
votes
Does allowing Ur-elements in $\sf ZF^*GC$ break bi-interpretability with $\sf ZF^*GC$?
Joel Hamkins has detected a gap in the proposed proof below. At the moment I do not know of another strategy to the answer the question, but I am leaving it below as signpost to others who might fall …
- 17.7k
9
votes
Reference for precise form of Mostowski: “sets correspond to rooted, well-founded, extension...
To my knowledge the basic idea of the correspondence between sets and graphs/trees of the type(s) specified in the question goes back to the work of Gödel (see Note 1 of his monograph on the consisten …
- 17.7k
6
votes
Accepted
Is every countable model of ZFC a subset of some parameter free definable pointwise-definabl...
The answer is in the positive. Since by the completeness theorem the existence of a countable model of ZFC is equivalent to Con(ZFC), the argument below works with the assumption that ZFC + Con(ZFC) …
- 17.7k
14
votes
Accepted
The existence of definable subsets of finite sets in NBG
The answer is in the negative. Let $\mathcal{M}$ be an $\omega$-nonstandard model of ZF, and $\mathfrak{X}$ be the collection of parametrically definable subsets of $\mathcal{M}$. Let $I$ be the cut …
- 17.7k
21
votes
Accepted
A contradiction in the Set Theory of von Neumann–Bernays–Gödel?
The comments to the question, especially those by Emil Jeřábek and Joel Hamkins make it clear that the proposed inconsistency proof breaks down because the recursive construction carried out in the pr …
- 17.7k
7
votes
Dedekind-Peano axioms, but numbers have at most one successor
As pointed out by Emil Jeřábek (in the comment to Joel Hamkins' answer), the system you are referring to is known as "Peano Arithmetic with a top". It has been studied for several decades. A good plac …
- 17.7k
12
votes
Accepted
What theories are larger than the real closed field but still decidable?
Thanks to a 1958 paper by Abraham Robinson (whose impetus was a question of Alfred Tarski), an example of such a theory that properly extends RCF is the theory of the structure $(\mathbb{R},~+,~\cdot …
- 17.7k