All Questions
Tagged with theories-of-arithmetic proof-theory
44 questions
5
votes
1
answer
142
views
Mathematical strength of the statement "Heyting Arithmetic admits Markov's rule"
Consider the following theorem about Heyting arithmetic (HA):
For every arithmetical formula $\phi$ whose only free variable is $n$, if $\text{HA} \vdash \forall n. \phi \lor \lnot \phi$ and $\text{...
- 6,403
2
votes
0
answers
100
views
Realizing arithmetic hierarchy in algebraic number theory
Is it possible to realize arithmetic hierarchy in algebraic number theory?
For example, consider a $\Pi^0_4$ statement of the form $\forall x \exists y \forall z \exists w \phi(x,y,z,w)=0$ where $\phi$...
- 593
5
votes
0
answers
109
views
Computational complexity of arithmetic sentences over classical theories
Below, I use the term "tracker" rather than "realizer" since I'm not requiring the relevant objects to be computable.
Define the relation "$f$ tracks $\varphi$" for $f:\...
- 20.7k
4
votes
1
answer
193
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Further research on relevant realizability etc
I just read Dunn's 1979 paper Relevant Robinson's Arithmetic, and the end especially caught my interest. Following the surprising role of constant functions in collapsing "relevant Q with zero&...
- 20.7k
3
votes
1
answer
135
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$\Pi^0_1$ sentences modulo "schematic entailment"
Let $\mathfrak{P}$ be the preorder of $\Delta^0_0$ (= only bounded quantifiers) formulas with one free variable in the language of arithmetic, under the relation $\alpha(x)\le\beta(x)$ iff there is a ...
- 20.7k
4
votes
1
answer
155
views
Does this hierarchy of fragments of $I \Sigma_1$ collapse?
Does anyone know whether the following hierarchy of fragments of
$\mathrm{I} \Sigma_1$ (or rather
$\mathrm{I} \Pi_1$) collapses or not?
Let $\Sigma^b_n$ denote formulas in the language of arithmetic ...
4
votes
0
answers
162
views
Can this theory of dyadic rationals prove that multiplying by three is the same as summing thrice?
(This question arose from a discussion with Jade Vanadium about a theory of dyadic rationals.)
Let $T$ be the 2-sorted FOL theory with sorts $ℕ,ℚ$ and constant-symbols $0,1$ and binary function-...
- 2,912
15
votes
5
answers
3k
views
How is it possible for PA+¬Con(PA) to be consistent?
I'm having some trouble understanding how a certain first-order theory isn't just straight-up inconsistent.
Let $PA$ be the axioms of (first-order) Peano arithmetic and let $C$ be the following ...
- 167
7
votes
0
answers
110
views
How tightly are decidability and "induction-completeness" linked?
It is known that there are a number of expansions of the structure $\mathfrak{N}:=(\mathbb{N};+)$ which are decidable (= have computable theories); one such example is the expansion by a predicate ...
- 20.7k
9
votes
0
answers
210
views
Is there an Arithmetized Completeness theorem for intuitionistic theories?
For classical theories, Henkin's completeness proof can be arithmetized. This leads to the result that for classical theories $T$ and $S$ if $\sigma$ is a formula enumerating $S$ in $T$ then $S \leq T ...
- 481
3
votes
0
answers
210
views
Self-referential Quinean proof of Löb's Theorem
Given its relevance for Open-source game theory, Dr. Andrew Critch asks the following about provability logic:
We conjecture that Löb’s Theorem can be proven without the use of the
modal fixed point $...
- 421
34
votes
2
answers
2k
views
What is the logical status of the sentence combining the ideas of Löb and Rosser, "this sentence is provable before any proof of its negation"?
Logicians are familiar with the variety of self-referential sentences expressible in the language of arithmetic:
The Gödel sentence, "this sentence is not provable", which indeed is not ...
- 237k
11
votes
0
answers
476
views
Which sentences are "irreducibly" self-referential over $\mathsf{PA}$?
Previously asked at MSE. Below, all sentences/formulas are in the language of arithmetic, and for simplicity we conflate numbers with numerals and sentences with Godel numbers.
Say that a sentence $\...
- 20.7k
7
votes
1
answer
358
views
Proving short consistency: can we do better than brute force search?
This is a minor variation of a question originally asked on MSE by user779130 and bountied by me, without success. Throughout, "length" refers to the number of symbols, not lines, in a proof....
- 20.7k
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
4
votes
1
answer
377
views
Does ACA prove categoricity of the reals?
$\def\f#1{\text{#1}}$Does $\f{ACA}$ prove that any two internally complete ordered fields are isomorphic?
Here internal completeness is expressed roughly as "every sequence of reals with an upper ...
- 2,912
9
votes
0
answers
204
views
Reverse mathematics of Noetherian rings over $\mathbb{Q}$
Take the Hilbert Basis Theorem over the rational numbers in this form in the language of Second Order Arithmetic: For every $n\in N$ every ideal of the polynomial ring $\mathbb{Q}[x_1,\dots,x_n]$ is ...
- 11.2k
10
votes
1
answer
630
views
Is $\mathsf{R}$ axiomatizable by finitely many schemes?
Recall that $\mathsf{R}$ is the theory of arithmetic consisting of the quantifier-free theory of $(\mathbb{N};+,\times,0,1,<)$ together with, for each $k\in\mathbb{N}$, the sentence $$\forall x[(\...
- 20.7k
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
4
votes
0
answers
203
views
The Return of Graham Arithmetics: adding induction up to $g_{64}$
In my previous question The inconsistency of Graham Arithmetics plus $ \forall n, n < g_{64}$, I introduced an extension of Robinson Arithmetics with the recursive definition of Tetraction, a small ...
- 7,853
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
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
6
votes
0
answers
422
views
What is proof-theoretic ordinal of weak first-order arithmetic?
According to Wikipedia(https://en.wikipedia.org/wiki/Ordinal_analysis) and nlab(https://ncatlab.org/nlab/show/ordinal+analysis), a proof-theoretic ordinal of $\mathsf{PRA}$ is $\omega^\omega$.
...
- 178
1
vote
1
answer
466
views
What does "can almost be proven in PA" mean regarding Theorem 2 of Timothy Chow's expository article, "The Consistency of Arithmetic"?
In his expository article, "The Consistency of Arithmetic" (MSN), Prof. Chow has the following theorems:
Theorem 1. If $a_1, a_2, a_3,\dotsc$ is a sequence of ordinals and $a_i \ge a_j$ whenever $...
- 6,132
7
votes
1
answer
246
views
Independent/Easy fraction of sentences over PA
Let $S(n)$ be the set of all sentences over PA of length at most $n$ (counting the quantifier symbols, boolean connectives, arithmetic operations and constants, and counting each variable as length $1$...
- 2,912
8
votes
1
answer
535
views
Is ZFC+(negation of a large cardinal axiom) arithmetically sound?
My knowledge in set theory is very limited, so I apologize if this question is naive or trivial:
Let $A$ to be a large cardinal axiom. $T=ZFC+\neg A$ is a consistent theory. My question is:
Question ...
- 870
4
votes
0
answers
292
views
the strength of saying "each sentence of true arithmetic has a recursive proof"
Let $PA_{\omega}$ be just like $PA$ except that $PA_{\omega}$-proofs can use any number of applications of the recursive $\omega$-rule.
The recursive $\omega$-rule allows the following:
For each ...
- 449
4
votes
1
answer
480
views
$f_{\epsilon_0}$ and provably total functions in $PA$
A total recursive function $f(x)$ is provably total in $PA$ if there's some formula $\phi(x,y)$ such that
$f(x)=y \iff PA\vdash \phi(x,y)$ and
$PA\vdash \forall x \exists y \phi(x,y)$
I know (not in ...
- 2,619
1
vote
1
answer
186
views
What is difference between length of proof and length of its presentation in Peano Arithmetic?
In this paper http://www.sciencedirect.com/science/article/pii/0304397584901117 page $19$ or $29$ it seems to imply there is a difference between length of proof and length of its presentation in ...
- 13.9k
3
votes
0
answers
198
views
What is the known weakest axiom system has Löb's derivability conditions?
We know that Peano Arithmetic satisfies Löb's derivability conditions, which is required in the proof of Gödel's 2nd incompleteness theorem. Is this the best result? If not, is there any known weaker ...
- 779
3
votes
0
answers
187
views
Reducing Consistency of $PA$ [closed]
By godel translation consistency of $PA$ is equivalent to consistency of $HA$.
I want to know any similar theorems for $PA$.
1.What is the minimal theory $T\subsetneq PA$ such that the proof of $PA\...
- 1,689
13
votes
3
answers
1k
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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
1
vote
2
answers
793
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 ...
- 1,112
8
votes
2
answers
560
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 ...
7
votes
1
answer
705
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 ...
- 777
2
votes
1
answer
347
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 ...
- 11.2k
11
votes
1
answer
433
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 ...
- 11.2k
12
votes
1
answer
976
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 ...
- 11.2k
9
votes
1
answer
1k
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 ...
2
votes
0
answers
84
views
Seeking name for an order raising operator in Higher Order Arithmetic.
Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of ...
- 11.2k
1
vote
3
answers
996
views
Applicability of Deduction theorem to Primitive recursive arithmetic [closed]
Hello. I already asked the question here. The main point is that I tried to prove in Primitive recursive arithmetic (PRA) the totality of the Ackerman function, and I found, that the single thing ...
- 15
4
votes
1
answer
876
views
Derivability conditions for Robinson arithmetic
Two pieces of hearsay I have encountered about Robinson's Q:
Q fails to satisfy the Löb derivability conditions;
Pudlák criticised the Löb derivability conditions and suggested rival, weaker ...
- 2,283
12
votes
1
answer
1k
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How to locate the paper that established Robinson Arithmetic?
If I'm not mistaken, it was in his seminal paper “An Essentially Undecidable Axiom System”, published in
Proceedings of the International Congress of Mathematicians (1950), 1952:729–730,
where R.M. ...
- 2,992
13
votes
3
answers
1k
views
Reducing ACA₀ proof to First Order PA
According to the Wikipedia ACA0 is a conservative extension of First Order logic + PA.
http://en.wikipedia.org/wiki/Reverse_Mathematics
First of all I have a few questions about the proof:
a - What ...
- 1,659