**2**

votes

**0**answers

10 views

### Proving moduli of uniform continuity in RCA_0

Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ...

**3**

votes

**3**answers

206 views

### Semantic reflection

Let $\ulcorner \cdot \urcorner$ be a fixed encoding of formulas by numbers, e.g.
let $\ulcorner \varphi \urcorner$ denote the Godel number of $\varphi$.
Let $T$ be a first-order arithmetic theory, ...

**8**

votes

**1**answer

343 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 ...

**11**

votes

**1**answer

235 views

### Can there be computable non-standard models of PA in a weaker sense?

By Tennenbaum's theorem, in the usual sense of computability for models,
neither addition nor multiplication can be computable in a non-standard model of PA.
Weak version:
Can addition or ...

**56**

votes

**4**answers

7k views

### Nelson's program to show inconsistency of ZF

At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel ...

**2**

votes

**1**answer

89 views

### What are the adequacy conditions for Rosser Provability?

Famously, Rosser introduced a provability predicate $\pi[A]$ that holds iff $\exists x(xP[A]\wedge\forall y(y\le x\to\lnot yP[\lnot A]))$.
Supposing $PA$ is consistent, what are the adequacy ...

**5**

votes

**0**answers

97 views

### Lascar strong types in fragments of arithmetic

Are Lascar strong types (definition below) in models of fragments of arithmetic always type definable? (They trivially are, in models of full induction.)
Definition Given a saturated model ${\cal M}$ ...

**-1**

votes

**1**answer

125 views

### Non-standard naturals and goodstein sequences [closed]

By the Kirby–Paris theorem, Goodstein's theorem is independent of Peano arithmetic (PA). Therefore there are non-standard models in which every Goodstein sequence terminates. However, Tennenbaum's ...

**4**

votes

**1**answer

181 views

### Induction and nonstandard halting times of standard machines

For a nonstandard model of enough arithmetic - say, $\mathcal{N}\models I\Sigma_1$ - we can define the set of halting times of standard machines relative to $\mathcal{N}$: ...

**2**

votes

**1**answer

113 views

### PA proves that functions are total

Is there a total recursive function $f:N \to N$ such that for no $\Sigma_1$ formula $\phi(x,y)$ which defines it (i.e., defines its graph), is it true that PA proves that "$\phi$ defines a total ...

**8**

votes

**2**answers

939 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

353 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 $\textsf{PA}$ ...

**12**

votes

**1**answer

419 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

240 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 ...

**6**

votes

**3**answers

402 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 ...

**14**

votes

**2**answers

937 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

137 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

162 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 ...

**8**

votes

**1**answer

638 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

137 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 ...

**6**

votes

**1**answer

663 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 ...

**7**

votes

**1**answer

244 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 ...

**10**

votes

**10**answers

4k views

### Is PA consistent? do we know it?

1) (By Goedel's) One can not prove, in PA, a formula that can be interpreted to express the consistency of PA. (Hopefully I said it right. Specialists correct me, please).
2) There are proofs ...

**12**

votes

**4**answers

563 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

303 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

275 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

155 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 ...

**28**

votes

**3**answers

2k views

### Even XOR Odd Infinities?

Modular Arithmetic (MA) has the same axioms as first order Peano Arithmetic (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
...

**8**

votes

**1**answer

393 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

284 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 ...

**27**

votes

**1**answer

459 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 ...

**12**

votes

**1**answer

437 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

205 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 ...

**6**

votes

**1**answer

307 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 ...

**4**

votes

**2**answers

372 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 ...

**1**

vote

**0**answers

71 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

463 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

388 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

251 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

137 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

358 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

278 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

**2**answers

722 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 ...

**15**

votes

**4**answers

2k views

### What is induction up to epsilon_0?

This is a question asked out of curiosity, and because I can't understand the wikipedia page.
I have often been told that PA cannot prove the validity of induction up to $\epsilon\_0$, which has been ...

**3**

votes

**3**answers

273 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

248 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 ...

**12**

votes

**1**answer

310 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 ...

**13**

votes

**6**answers

1k views

### Non-constructive proofs of decidability?

Are there examples of sets of natural numbers that are proven to be decidable but by non-constructive proofs only?

**7**

votes

**3**answers

526 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 ...

**8**

votes

**4**answers

2k views

### Incompleteness and nonstandard models of arithmetic

The following are a collection of doubts, some of which shall have concrete answers while others may have not. Any kind of help will be welcome.
Reading Peter Smith's "Gödel Without (Too Many) ...