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, ...

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17
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2answers
1k views

Analogues of Primitive Recursive Functions

Let $\mathbf{A}$ be an admissible set (possibly with urelements). I am wondering if there is some good notion of "primitive recursive arithmetic" relative to $\mathbf{A}$. More precisely, I would like ...
61
votes
39answers
12k views

nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because ...
0
votes
0answers
115 views

local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
6
votes
1answer
183 views

Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. Theorem The normal subgroups of $S_\infty$ are ...
29
votes
2answers
3k views

Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...
5
votes
1answer
365 views

Infinite blue eyed islanders puzzle

Can the well known blue eyed islanders puzzle be extended to an infinite number of islanders? In that puzzle, a set of $k$ islanders, each with either blue eyes or non-blue eyes, each knows the color ...
-1
votes
0answers
50 views

Ddifference between deduct and deduce [on hold]

In the context of logic, the term "deduction" was used as a way of thinking. I think, apparently, the verb of "deduction" should be "deduct", hence it is natural and reasonable to use the term ...
1
vote
1answer
144 views

Interpreting peano arithmetic without parameters

I will accept an answer in the form of references to the literature about my question as well as any other information. I am quite ignorant of the area and that will be clear from my question. I ...
2
votes
0answers
66 views

Are Braid Groups with Finitely many Generators NIP?

I am curious what braid groups (strings in $\mathbb{R}^3$) are NIP. Consider the following: Let $B_\mathbb{N}$ be braid group with "braids" indexed by the natural numbers (alternatively, the direct ...
1
vote
1answer
216 views

Sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$

Consider sets of Vitali's type in models of $\mathsf{ZF}+\mathsf{GCH}$ where $V \neq L$. Are there sets of Vitali's type in both $L$ and $V \backslash L$? If so, is there any way one can distinguish ...
2
votes
1answer
164 views

What is the extension of the truth-table degrees to Baire Space called?

Recall that for sets $A, B \in 2^\omega$ that we say $A \leq_{tt} B$ if there is a total Turing functional $F \colon 2^\omega \to 2^\omega$ such that $F(B)=A$. This is called truth-table ...
8
votes
0answers
149 views

Turing degree of finding independent formulas

In this Paper of D. Isaacson, it is proved that the true arithmetic($Th(\mathbb{N}$)) is the only $\omega$-consistent and complete extension of $Q$ (Robinson's arithmetic). This, together the fact ...
2
votes
2answers
349 views

What are the sense and reference of the propositions $R \notin R$, $R \in R$, where $R=\{x \mid x \notin x\}$ in Frege's Grundgesetze?

In the paper, Aldo Antonelli and Robert May, Frege's new science, Notre Dame J. Form. Log. 41 (2000), no. 3, 242–-270, MR 1943495. the authors give the following quote of Frege, from his paper ...
3
votes
1answer
233 views

Representation of meager sets in Cohen extensions

Let $M$ be a transitive model of ZFC and $c\in {}^\omega2$ a Cohen real over $M$. Let $A$ be a meager Borel subset of $^\omega2$ in $M[c]$. I would like to prove that there exists a meager Borel set ...
30
votes
2answers
3k views

“Transitivity” of the Stone-Cech compactification

Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \setminus \mathbb{N}$ be two non-principal elements of this ...
9
votes
5answers
306 views

Probability theory without deductive closure

Human knowledge is not deductively closed. Uncertainty can arise from that just as much as from lack of brute facts. (When a Harvard graduate was reported to have thought that the earth is farther ...
2
votes
6answers
2k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, ...
4
votes
0answers
175 views

Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations. This question assumes familiarity with combinatorial cardinal characteristics of ...
4
votes
0answers
91 views

Automorphism group of a structure without the SAP

A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property. It is ...
20
votes
2answers
2k views

Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...
5
votes
1answer
82 views

Complexity of integer programming with added predicates

A classical theorem in Integer Programming by Lenstra says that any integer system $$A x \le b$$ can be solved in polynomial time, where $A \in \mathbb{Z}^{m \times n}, x \in \mathbb{Z}^n, b \in ...
6
votes
0answers
117 views

Examples of analytic $\mathcal{I}$-mad families

If $\mathcal{I}$ is an ideal (proper and containing the finite sets) on $\omega$, call a family of subsets $\mathcal{A}\subseteq[\omega]^\omega$ $\mathcal{I}$-almost disjoint if for all distinct ...
8
votes
0answers
179 views

Inner models and strongly compact cardinals

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly ...
4
votes
0answers
138 views

Spectrum Problem for Higher-Order Logic

Definitions. Given a sentence $\varphi$ of $n$th-order logic, we define the spectrum of $\varphi$ to be the set of cardinalities of finite structures that satisfy $\varphi$. A set $X\subseteq\mathbb ...
1
vote
0answers
39 views

Quantifier elimination of pp-subgroups of modules

This is a model-theoretic questions. Let $M$ be a $R$-module. Our language will be the standard language of modules, i.e. the language of abelian groups together with an unary function symbol for ...
4
votes
1answer
141 views

Expressive power of $\omega$-order logic

According to the article Second-order and Higher-order Logic from the Stanford Encyclopedia of Philosophy, there is no need to stop at second-order logic; one can keep going. [...] we can allow ...
10
votes
1answer
259 views

Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa ...
14
votes
1answer
464 views

Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?

The question is: if I assert in ZF that there exists a Reinhardt cardinal, do I really get a theory of higher consistency strength than when I assert in ZFC that there exists an I0 cardinal (the ...
10
votes
1answer
321 views

Uniform elimination of imaginaries

Does the following principle follow from uniform elimination of imaginaries? For every formula $\varphi(x;y)$ there is a formula $\vartheta(x;z)$ such that $$\forall y\;\exists^{=1}z\;\forall ...
21
votes
13answers
2k views

Are there any good nonconstructive “existential metatheorems”?

Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is ...
26
votes
4answers
2k views

Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...
11
votes
4answers
1k views

How fast can the base-bumping function in Goodstein's theorem grow?

In the usual presentation of Goodstein's theorem, the base is bumped up by the "add 1" function. Does the theorem still hold when we replace this function by a fast-growing one (e.g. Ackermann or busy ...
9
votes
4answers
3k views

why Skolemization?

Can someone explain what is the reason for using Skolemization, I clearly understand that its the removal of existencial quantifiers, but whats the use? why is keeping existencial quantifiers not ...
42
votes
15answers
6k views

Abstract Thought vs Calculation

Jeremy Avigad and Erich Reck in their remarkable historical paper "Clarifying the nature of the infinite: the development of metamathematics and proof theory" claim that one of the factors of becoming ...
1
vote
1answer
131 views

$\mathcal A\equiv\mathcal B\implies \mathcal A\cong\mathcal B$ for finite $\mathcal L$-structures where $\mathcal L$ is an infinite signature

Let $\mathcal L$ be an infinite signature and $\mathcal A$, $\mathcal B$ two finite $\mathcal L$-structures such that for each first-order $\mathcal L$-sentence $\varphi$, $$\mathcal ...
61
votes
27answers
8k views

Sexy vacuity …

I included this footnote in a paper in which I mentioned that the number of partitions of the empty set is 1 (every member of any partition is a non-empty set, and of course every member of the empty ...
4
votes
1answer
214 views

Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ “get larger and larger”?

Is the following statement consistent in $\mathsf{ZFC}$? For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have ...
7
votes
1answer
229 views

Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question. For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...
19
votes
2answers
1k views

What is the largest Laver table which has been computed?

Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$ $$a* (b* c) = (a* b) * (a * c).$$ This is the $n$th Laver table ...
10
votes
1answer
351 views

What Turing degree is this function?

Over at http://www.scottaaronson.com/blog/?p=2725#comment-1089004 we had a discussion of intermediate Turing degrees. The following function came up: Take Chaitin’s constant, and rearrange its ...
11
votes
1answer
397 views

Changing cofinalities above supercompact cardinals

Question. Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in ...
-8
votes
1answer
349 views

Missing Axiom: There are no other axioms. Leads to a proof of CH within ZFC [closed]

I have come to the conclusion that there is often an implied axiom: There are no other axioms. Failure to explicitly state this axiom and to consider its consequences can result in the misleading ...
0
votes
1answer
152 views

Can epsilon-induction be derived from the transitive closure operator?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations, if we do not have the Foundation Axiom. The ...
19
votes
1answer
1k views

Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
1
vote
1answer
230 views

Natural number properties as uninterpreted functions in first order logic

Can we express the following property of natural numbers as FOL. The property given below is only indicative, I am more interested in knowing how the concepts such as "infinitely many X exists for so ...
8
votes
2answers
296 views

“Clubiness” of projective sets of ordinals

I'm sure this is just my google-fu failing me, but: what are sufficient, non-overkill large cardinal axioms which guarantee "Every (boldface) $\Pi^1_n$ set of (real codes for) countable ordinals ...
3
votes
1answer
698 views

Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus? Is $\lambda x$ the same as $\tau_x$?

Is there any relationship between Bourbaki's Epsilon Calculus and Lambda Calculus ? Whether $\lambda {x}$ is same as $\tau _{x}$ ? Are the rules of Meta-Mathematics (Criteria of Substitution, ...
28
votes
15answers
2k views

Objects which can't be defined without making choices but which end up independent of the choice

It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure. And sometimes ...
11
votes
1answer
334 views

Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact ...
2
votes
2answers
178 views

Henkin semantics for Second-order Logic

I know that the natural numbers can be categorically characterized in second-order logic with the standard semantics. However, I could not find an example of a non-standard Henkin structure (one that ...