All Questions
Tagged with peano-arithmetic or theories-of-arithmetic
334 questions
26
votes
3
answers
7k
views
Presburger Arithmetic
Presburger arithmetic apparently proves its own consistency. Does anyone have a reference to an exposition of this? It's not clear to me how to encode the statement "Presburger arithmetic is ...
- 464
39
votes
3
answers
3k
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 ...
5
votes
1
answer
283
views
Which arithmetical sentences have no counterexamples in the sense of Kreisel?
It is a well-known fact that given a first-order sentence $\psi$ in prenex normal form $\forall x_1 \exists y_1 \forall x_2 \exists y_2 \dots \forall x_n \exists y_n \theta(x_1,\dots,x_n,y_1,\dots,y_n)...
- 13.2k
14
votes
3
answers
2k
views
Which recursively-defined predicates can be expressed in Presburger Arithmetic?
In Presburger Arithmetic there is no predicate that can express divisibility, else Presburger Arithmetic would be as expressive as Peano Arithmetic. Divisibility can be defined recursively, for ...
13
votes
1
answer
437
views
What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?
On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) ...
- 13.2k
1
vote
0
answers
117
views
Can this type theory interpret second order arithmetic?
Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
- 11.3k
8
votes
1
answer
574
views
Iterated Gentzen: or, a Sith objection to the proof of consistency of PA
$\DeclareMathOperator\PRA{PRA}\DeclareMathOperator\WF{WF}\DeclareMathOperator\Con{Con}\DeclareMathOperator\PA{PA}$Preamble: In the year … in a galaxy far far away, a nasty Sith named Darth Dubious (...
- 7,853
9
votes
1
answer
494
views
What can $I\Delta_0$ prove?
What combinatorial and number-theoretic propositions can $I\Delta_0$ prove? Obviously there are an infinitude of them, but what are some well known theorems that can be proved in $I\Delta_0$, if any?
- 675
16
votes
3
answers
19k
views
Non-computable but easily described arithmetical functions
I have read about the existence of functions of the kind described in the title in several places, but never seen an instance of them. Sorry if this is too much an elementary question to be posted ...
- 1,465
8
votes
1
answer
1k
views
Can the "real" Peano Arithmetic be inconsistent?
Assuming $\text{PA}$ is consistent. Then $\text{PA} + \neg\text{Con}(\text{PA})$ is consistent and have a model, say $M$. We know $M$ must be nonstandard, in which case, there is a nonstandard proof ...
- 779
0
votes
0
answers
152
views
What is the strength of allowing multiple predecessor numbers?
If we have a theory of numbers, pairs of numbers, and sets of those, and axiomatize that the relation $<$ on numbers is both extensional and well founded, then this theory would prove all PA axioms ...
- 11.3k
11
votes
1
answer
400
views
What is the Turing degree of the monadic theory of the real line?
The monadic theory of the real line is the set of all sentences in the monadic second-order language of order which are true in $\mathbb{R}$. In this 1982 paper, Gurevich and Shelah show that true ...
- 4,597
6
votes
2
answers
436
views
Interpreting proper elementarily equivalent end extensions?
Is there a tuple of parameter-free formulas $\Phi$ and a nonstandard $M\models PA$ such that $\Phi^M\models PA$, the induced $M$-definable initial segment embedding $j_\Phi^M:M\rightarrow\Phi^M$ is ...
- 20.7k
17
votes
1
answer
2k
views
What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
- 6,403
30
votes
2
answers
3k
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 $\exists x(Sx = 0)$.
(http://en.wikipedia.org/wiki/Peano_axioms#First-...
- 687
0
votes
1
answer
150
views
Reflection schema
Peano Arithmetic consists of axioms $P_1, P_2, \ldots P_7$ plus first order classical logic. Let us call this theory $T$. This theory has its unprovable Gödel’s sentence $G$ such that
$$
G\...
- 1
57
votes
2
answers
7k
views
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
- 30.3k
5
votes
2
answers
2k
views
How many models of Peano arithmetic are isomorphic to the standard model and how many models of Peano arithmetic are non-standard?
I am currently writing a paper on non-standard models of Peano arithmetic and I am having trouble finding references or information that discuss the relative sizes of how many models of Peano ...
- 1,441
16
votes
2
answers
714
views
Is (Z,+,0,1,P2,P3) decidable?
Is Presburger arithmetic, augmented with two unary predicates P2, P3, for powers of 2 and powers of 3 respectively, decidable?
I know that adding just one of P2, P3 to Presburger keeps it decidable, ...
- 428
4
votes
0
answers
198
views
Is there a simple proof of consistency of EA?
Let $\mathsf{EA}+\mathsf{CE}$ be elementary arithmetic with cut elimination theorem. Is there a simple (1-)consistency proof of $\mathsf{EA}$ over $\mathsf{EA}+\mathsf{CE}$? I think that a naïve ...
- 178
18
votes
3
answers
1k
views
Computable nonstandard models for weak systems of arithmetic
By Tennenbaum's theorem, PA itself does not have any computable nonstandard models. The integer polynomials which are 0 or have a positive leading coefficient form a computable nonstandard model of ...
user5810
2
votes
1
answer
147
views
Representation of the equality relation between hereditarily finite sets in weak set theories
Consider General Set Theory ($ \mathsf { GST } $) axiomatized by the following.
Axiom of Extensionality: The sets $ x $ and $ y $ are the same set if they have the same members:
$$ \forall x \forall ...
- 346
3
votes
0
answers
165
views
Why is the proof of decidability of arithmetic (Theorem 2.1) in Hamkins & Lewis (2000) enough?
Recently, I was reading the paper "Infinite Time Turing Machines" by Hamkins & Lewis. And one of the first theorems (Theorem 2.1) is about decidability of arithmetic.
The proof is quite ...
- 31
7
votes
0
answers
284
views
Generic behavior of "polynomialish" models of $\mathsf{Q}$
(This question was originally asked and bountied at MSE - with different notation, some more explicit arguments, and topology in place of forcing.)
Suppose $\mathcal{R}=(R_i)_{i\in\mathbb{N}}$ is a ...
- 20.7k
27
votes
5
answers
4k
views
What is induction up to $\varepsilon_0$?
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 $\varepsilon_0$, which has ...
- 156k
17
votes
3
answers
3k
views
Gödel's Incompleteness Theorem and the complexity of arithmetic
In How complicated can structures be? Jouko Väänänen says:
The guiding result of mathematical logic is the Incompleteness Theorem of Gödel,
which says that the logical structure of number theory ...
- 9,730
13
votes
3
answers
1k
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 ...
- 28.2k
0
votes
1
answer
270
views
What is the smallest countable limit ordinal in which 'lost melodies' occur
The question is in the title. This question is in response to the following paragraph found at the end of Prof. Hamkins' answer to my MathOverflow question, Are ITTM's necessary to compute Turing's &...
- 6,132
8
votes
2
answers
2k
views
Axiom to exclude nonstandard natural numbers
In Peano Arithmetic, the induction axiom states that there is no proper subset of the natural numbers that contains 0 and is closed under the successor function. This is intended to rule out the ...
- 295
5
votes
0
answers
318
views
$\Sigma_n$-complete sets in the Levy hierarchy
Recall that a set $A \subseteq \mathbb N$ is (many-one, Turing) $\Sigma_n$-complete if it's $\Sigma_n$ and any other $\Sigma_n$ set (many-one, Turing) reduces to it. This definition actually makes ...
- 1,882
2
votes
1
answer
198
views
Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?
In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that
$$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \...
- 798
34
votes
8
answers
8k
views
Arithmetic fixed point theorem
I want to understand the idea of the proof of the artihmetic fixed point theorem. The theorem is crucial in the proof of Gödel's first Incompletness theorem.
First some notation: We work in $NT$, the ...
- 63.1k
8
votes
3
answers
427
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 ...
- 1,403
22
votes
5
answers
1k
views
What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...
- 237k
0
votes
3
answers
1k
views
Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")
Consider the following note written by Gerhard Gentzen in early 1932, on the onset of his work on a consistency proof for arithmetic:
The axioms of arithmetic are obviously correct, and the ...
- 6,132
10
votes
1
answer
414
views
Concrete examples of statements not provable in PRA + $\epsilon_0$-induction that are provable in PA?
It is well-known that $\mathbf{PRA}$ plus $\epsilon_0$-induction on bounded formulas cannot prove all $\mathbf{PA}$ theorems (essentially because $I\Sigma_1$ plus $\epsilon_0$-induction on bounded ...
- 189
10
votes
1
answer
350
views
An internalized version of Tennenbaum's Theorem
Tennenbaum's celebrated 1959 theorem (see here for a reference) is certainly one of the key theorems in mathematical logic. Not so much for its proof, but because it helps "isolating" $N$ ...
- 7,853
5
votes
1
answer
271
views
Can a nonstandard model of $\mathsf{PA}$ be "$\Delta^1_1$-well-ordered?"
This was asked and bountied at MSE with no response:
My question is the following:
Is there a nonstandard model $\mathcal{M}\models\mathsf{PA}$ such that $\mathcal{M}$ has no $\Delta^1_1$-with-...
- 20.7k
8
votes
1
answer
283
views
Paris-Harrington principles parametrized by functions $f:\mathbb N \to \mathbb N$
Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \...
- 1,882
9
votes
1
answer
644
views
Gentzen's result on PA
The Wikipedia states that Gentzen proved that "in modern terms, the proof-theoretic ordinal of PA is $\varepsilon_0$." Further down, the article defines what the "proof theoretic ordinal" of a theory ...
- 1,174
5
votes
1
answer
393
views
Lob theorem for Robinson arithmetic
If i'm not wrong, the theory which Lob theorem applies to should be sufficiently strong, satisfying 3 "derivability" conditions, like PA.
$Q$ is the Robinson arithmetic.
I'm afraid $Q$, is ...
- 103
6
votes
0
answers
428
views
Proof of Tennenbaum's Theorem by McCarty
Tennenbaum's Theorem in its usual form states that for any countable non-standard model $M$ of PA there is no way to code the elements of $M$ as natural numbers such that either the addition or ...
- 211
6
votes
1
answer
727
views
What is the consistency strength of this theory?
Language: first-order logic
Primitives: $=, S, \in $ (the first denotes identity, the second denotes “is a successor of”, and the third denotes membership relation).
Axioms: those of identity ...
- 11.3k
4
votes
0
answers
553
views
Can Robinson arithmetic prove any interesting theorems?
The motivation for my question is I'm curious whether studying Robinson arithmetic can be fruitful in the same sense as studying group theory. Robinson arithmetic is so weak that there are many ...
- 675
2
votes
2
answers
436
views
Is there any reasonable non-regular Gödel numbering of the language of arithmetic?
Let $\mathcal{L}$ be the language of arithmetic given as follows:
$x::= {\sf v} \mid x'$
$t ::= x \mid 0 \mid {\sf S}t \mid (t+t) \mid (t\times t)$
$A ::= \bot \mid \top \mid t=t \mid \neg A \mid (A \...
2
votes
1
answer
275
views
Definability in countable nonstandard models of Peano arithmetic
I know that every element of $\mathbb{N}$ is definable the standard model of Peano Arithmetic. Does there exist a countable non-standard model of PA where the same is true?
7
votes
1
answer
489
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 $\Pi_1^1$...
- 71
42
votes
7
answers
3k
views
How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
This question is closely related to this one: Knuth's intuition that Goldbach might be unprovable. It stems from my ignorance about non-standard models of arithmetic. In a comment on the other ...
- 29k
3
votes
1
answer
148
views
Consistency and consistency strength of certain special cuts in $I\Delta_0$
Recall that $I\Delta_0$ is the theory in the language of arithmetic that consists of the axioms of $\mathsf{PA}$ with induction restricted to $\Delta_0$ formulas (i.e., formulas where all quantifiers ...
- 13.2k
12
votes
2
answers
1k
views
Trouble with models of PA and ZFC
I have a big trouble in my mind, here is my false reasoning:
The Goodstein's theorem is undecidable in (first order) Peano Arithmetic.
There exist a non standard model N of PA where the Goodstein's ...
- 129