**8**

votes

**2**answers

834 views

### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...

**4**

votes

**1**answer

288 views

### Does PA+Con(PA) entail the existence of non-standard models of PA?

Does $\textsf{PA}$+Con($\textsf{PA}$) entail the existence of non-standard models of $\textsf{PA}$?
Is there a reasonable way in which to code, inside $\textsf{PA}$, the statement that $\textsf{PA}$ ...

**11**

votes

**1**answer

318 views

### Is factorial definable using a $\Delta_0$ formula?

The factorial function is primitive recursive, and therefore definable by a $\Sigma_1$ formula.
Is it also definable by a $\Delta_0$ formula (i.e. bounded quantifiers)?
If not, why?

**7**

votes

**1**answer

223 views

### Is $ACA_0$ + `True Arithmetic exists' interpretable in $ACA$?

Maybe someone here can help me with a question concerning second-order arithmetic. Consider the system $ACA_T := ACA_0 + \exists X \forall x (x \in X \leftrightarrow T(x))$, where $T(x)$ is a ...

**13**

votes

**2**answers

823 views

### nonstandard models and mathematical theorems

Is there a first order statement about the natural numbers (not nonstandard analysis) such that the truth of the statement is easier to see in a nonstandard model? In other words, do nonstandard ...

**2**

votes

**1**answer

115 views

### Total formulae in a theory equivalent to $\Delta_0$ formulae in the theory?

Let a formula $\phi$ of the language of first-order Peano arithmetic be total in a theory Th that extends PA iff, for any $k_1, \dots, k_n \in \omega$, Th $\vdash \phi(\bar k_1, \dots, \bar k_n)$ or ...

**2**

votes

**1**answer

136 views

### Adding consistency statements to Peano arithmetic allows more instances of transfinite induction?

Consider the hierarchy given by $\cal S_0 =$ first-order Peano arithmetic, $\cal S_{\alpha+1}=\cal S_{\alpha} + Con(S_\alpha)$ (a consistency statement for $\cal S_\alpha$), and if $\alpha$ is a limit ...

**7**

votes

**1**answer

572 views

### Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following:
Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...

**3**

votes

**1**answer

130 views

### Formal systems needed to formalize relative independence results

We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they ...

**7**

votes

**1**answer

200 views

### Which ordinals can be proof-theoretic ordinals of a reasonable theory?

When talking to a friend recently he asked a question - are there any reasonable first-order theories which have proof theoretic ordinal equal to small or large Veblen ordinal? I have then extended ...

**6**

votes

**1**answer

621 views

### Does Nelson try to prove PA inconsistent directly?

Edward Nelson is known for his serious attempts to show that Peano axioms, and sometimes even weaker theories, are inconsistent. I wasn't able to find Nelson's papers anywhere, so I wanted to ask a ...

**10**

votes

**4**answers

524 views

### Boolean Valued Models of PA

O.K, a massively naive question. I've never really studied any non-standard models of PA before. I was just wondering if there's ever been any attempt to use the kind of Boolean valued model theory ...

**1**

vote

**2**answers

272 views

### An interpretation of not-Con(PA)

Edit After Andreas Blass answer below and comments below the original post I have changed it to accommodate posters' remarks. I hope it is clear and makes more sense now.
Let $\mathrm{PA}$ be the ...

**4**

votes

**1**answer

257 views

### “Is it possible to give a restricted set-theoretical definition of addition of natural numbers in terms of successor?” [Tarski]

In his paper "Restricted set-theoretical defintions in arithmetic" Raphael Robinson cites a problem posed by Tarski:
Is it possible to give a restricted set-theoretical
definition of addition of ...

**0**

votes

**1**answer

140 views

### Definability of arithmetic functions and relations

Motivation: Many "weak" arithmetic functions and/or relations ("relations" for short) are equivalent with relations explicitly definable by relations which were recursively defined by them beforehand ...

**8**

votes

**1**answer

371 views

### The (un)decidability of Robinson-Arithmetic-without-Multiplication?

I asked this over at math.stackexchange, and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so ...

**7**

votes

**1**answer

271 views

### The definition of < in Robinson's Q

I recently had to explain how the basic axioms in Simpson's Subsystems of Second Order Arithmetic were interpretable in Robinson's Q. Most of the axioms are actually the same, except that Simpson ...

**7**

votes

**1**answer

296 views

### Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...

**12**

votes

**1**answer

409 views

### Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...

**6**

votes

**2**answers

188 views

### Models of PRA/EFA with induction on $X$ but not $\omega^X$

As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...

**4**

votes

**2**answers

349 views

### Overspill in models of arithmetic

Assume that $M$ is a non-standard model of complete arithmetic, i.e. of the theory $Th(\mathbb{N})$. Suppose that $R$ and $S$ are proper cuts of $M$. (With a cut, I mean a subset of the universe of ...

**6**

votes

**1**answer

293 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**1**

vote

**0**answers

64 views

### First-order Peano Axioms and order-completeness of $\mathbb{N}$ [closed]

Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo.
Notation: We denote the system of first-order Peano Axioms (along with ...

**2**

votes

**3**answers

422 views

### Can a Decidable Theory Have Non-recursive Models?

Tennenbaums' theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA.
...

**1**

vote

**1**answer

360 views

### Transfinite induction vs induction in mathematics

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in ...

**2**

votes

**1**answer

244 views

### Elementary proof of bounds on factor polynomials

The question Getting a bound on the coefficients of the factor polynomial got very nice answers on Gelfond's theorem. But for work on proof theory of arithmetic I want a proof in arithmetic. The ...

**3**

votes

**0**answers

122 views

### Peano (Dedekind) categoricity

What is the smallest fragment of second order logic such that $Th(\mathbb{N})$ in that logic is categorical (only one model, namely natural numbers, up to isomorphism). For example, can we do this in ...

**0**

votes

**1**answer

334 views

### Recursive Non-standard Models of Modular Arithmetic? [closed]

Any algebraically closed field (ACF) is a model of Modular arithmetic (MA). (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists ...

**3**

votes

**2**answers

269 views

### Efficient representations of natural numbers via arithmetical expressions

A given natural number $n \in \mathbb{N}$ has many representations
as expressions mixing other natural numbers and the operators and punctuation symbols
$\{+,-,\times,/,\exp,(,)\}$, where '$\exp$' ...

**3**

votes

**3**answers

221 views

### Turing Functional and $\Sigma_1^0$-formulas in models of fragments of PA

In models of PA with restricted induction power (for example, only $I\Sigma_n$ is present), the failure of higher induction scheme is characterised by the existence of definable cuts (like $\Sigma_2$ ...

**4**

votes

**1**answer

243 views

### Interpreting the Galois theory of finite extensions of $\mathbb{Q}$ in PA

Any finite extension of the rationals, along with its Galois group, can be interpreted in Peano arithmetic by straightforward means. For a fixed bound $n$ in the degree this is uniform in the ...

**6**

votes

**3**answers

354 views

### Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...

**3**

votes

**2**answers

696 views

### Neither Even Nor Odd Natural Numbers

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. In Even XOR Odd Infinities? I asked if this statement ...

**7**

votes

**3**answers

484 views

### Elementary end extensions of models of Peano Arithmetic in uncountable languages

A well-known theorem of Mills asserts that there is a model of Peano Arithmetic $M$ in an uncountable language such that $M$ has no elementary end extension (e.e.e.). I ask whether every complete ...

**9**

votes

**3**answers

277 views

### Uncountable model of bounded arithmetic with an elementary end extension

Theorem 1.53 (3) in page 227 of Hajek and Pudlak's book, Metamathematics of First-Order Arithmetic, says:
Theorem. If $M$ is a countable model of $I\Delta_{0}$ such that $M$ has a proper elementary ...

**6**

votes

**1**answer

165 views

### A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$

The question has relevance for constructing Scott sets with certain extra desirable properties.
Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...

**8**

votes

**3**answers

327 views

### If an oracle Turing machine halts with every infinite arithmetic oracle, can it fail to halt with some non-arithmetic oracle?

Let $e$ be an index of an oracle Turing machine program and $k$ be some natural number. Let us say that a subset of $\mathbb N$ is arithmetic if it is definable in the model $\langle \mathbb ...

**7**

votes

**1**answer

620 views

### Has Ribet's theorem been proved using only finite powers of primes?

Ribet proved the Serre epsilon conjecture using $p$-adic Galois representations (http://math.berkeley.edu/~ribet/Articles/invent_100.pdf). Can someone show how to replace all use of $p$-adics in ...

**30**

votes

**3**answers

2k views

### Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?

I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...

**1**

vote

**1**answer

209 views

### Infinite board games: sentences about

As a unified approach if we have an ( read any) infinite board game described as $\mathcal{G}$ using a particular axiom set A..
can a sentence be devised in A which automatically answers the basic ...

**3**

votes

**2**answers

271 views

### Reference request: Minimal Axiomatizations of PA over (+,x,<=).

Many years ago, when I was still a high school student, I came up with a certain first-order axiomatization of PA over the signature (+, x, ≤). Out of nostalgia, I've decided to clean up what I ...

**38**

votes

**2**answers

2k views

### What arithmetic information is contained in the algebraic K-theory of the integers

I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...

**26**

votes

**1**answer

433 views

### Decidability of equality of expressions built using 1,+,-,*,/,^

Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in ...

**11**

votes

**1**answer

397 views

### Does any lower bound on proofs of FLT improve Shepherdson 1965?

In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl ...

**12**

votes

**1**answer

282 views

### What metatheory proves $\mathsf{ACA}_0$ conservative over PA?

Simpson's book shows $\mathsf{ACA}_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}_0$ being conservative over PA is ...

**1**

vote

**1**answer

846 views

### Are there non-commutative models of arithmetic which have a prime number structure?

Peano Arithmetic (PA) models have a prime number structure and commutativity of addition and multiplication. Presburger arithmetic (PrA) models of arithmetic have addition without multiplication and ...

**4**

votes

**1**answer

266 views

### ERA, PRA, PA, transfinite induction and equivalences

I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.
For instance I'm ...

**0**

votes

**2**answers

812 views

### Gödel, Escher, Bach: b is a power of 10. [closed]

I’d like to verify if my formula correctly expresses that a number is a power of $ 10 $, using the $ \sf{TNT} $ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal ...

**2**

votes

**3**answers

2k views

### Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom:
$$\forall x \forall y \forall ...

**0**

votes

**0**answers

285 views

### Definitions for Oddness

In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 ...