All Questions
Tagged with set-theory theories-of-arithmetic
79 questions
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About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
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-4
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0
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133
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Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
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-4
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1
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To which arithmetic\set theory this theory is bi-interpretable?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
$ \textbf{Axioms:}$
$ \textbf{Order:} \ x < y < z \to x < z $
$ \textbf{...
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4
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Is this theory synonymous with PA?
Language: Mono-sorted first order logic with equality.
Extralogical Primitives: $<, \in$
Define: $x \leq y \iff x < y \lor x=y$
$\textbf{Well ordering: }\\\textit{Transitive:} \ x < y \land ...
- 11.3k
8
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2
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2k
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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 ...
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30
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2
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3k
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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-...
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19
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3
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Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
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1
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1
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Natural functions outside $\sf PA$?
Can theories stronger than $\sf PA$ manage to define functions from the naturals to the naturals, that $\sf PA$ cannot? If so, what are examples of those functions?
- 11.3k
1
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1
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Can PA define functions related to higher theories?
Working in $\sf ZFC$ we can define a function $F$ from naturals to naturals such that $F(0) = \, ^\circ\ulcorner r({\sf Z}) \urcorner$, where $r({\sf Z})$ is the Rosser sentence of Zermelo set theory $...
- 11.3k
1
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2
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Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
I've relatively recently learned about Goodstein's Theorem and its unprovability in Peano arithmetic (the Kirby-Paris Theorem). I do not have any real knowledge of formal logic; but I think I've seen ...
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Is possibile to define transfinite sum and product recursively? [closed]
On mathstackexchange a few days ago I published the following question where I asked about "transfinite" sum and products but actually nobody answered or gave an opinion with a comment: thus ...
4
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1
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515
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Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
- 2,167
5
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2
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432
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Models of second-order arithmetic closed under relative constructibility
I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
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1
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Would this alteration safeguard the resulting theory from inconsistency?
If we replace "Emergence" axiom in the theory $T$ presented at posting "What is the set theory synonymous with this order-set theory?" with the following axiom, call the resulting ...
- 11.3k
3
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1
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Would this alteration of $T$ affect its synonymy with PA?
If we replace the single axiom "Emergence" in the axiomatic system $T$ presented at posting [What is the set theory synonymous with this order-set theory], by the following schema. Would the ...
- 11.3k
1
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1
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313
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What is the set theory synonymous with this order-set theory?
Let $ T$ be a theory written in Mono-sorted first order logic with equality, with extralogical primitives: $<, \in$.
Define: $x \leq y \iff x < y \lor x=y$
Axioms:
$\textbf{Well ordering: }\\\...
- 11.3k
10
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2
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600
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Is diamond consistent with 2nd order PA?
If $T$ is a theorem of ZF which says something only about reals, then one may want to prove $T$ using a theory like 2nd order PA or related theories like ZFC$^-$ or GBC$^-$ (minus accounts for the ...
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5
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In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ is consistent?
It seems common amongst logicians to think of "truth" as being relative to a particular structure. Consider, for instance, the first-order theory of groups. The sentence $\forall x\forall y(...
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Defining the standard model of PA so that a space alien could understand
First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on ...
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6
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407
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Can Set Theory be turned into Infinite Arithmetic?
The following system I'd label as "Infinite Arithmetic" is simply an endeavor to extend second order arithmetic to the infinite ordinal world, and extending with it the representation of ...
- 11.3k
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Is set theory interpretable in infinite primitive recursive arithmetic?
In A Formalization of the Theory of Ordinal Numbers, Takeuti interprets $\sf ZFC$ in a first order theory extending first order arithmetic to the infinite ordinal realm, while at the same time ...
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Can set theory be interpreted in infinite arithmetic?
Is the following system of infinite arithmetic consistent?
If so, can it interpret $\sf ZFC$?
Language: first order logic
Primitives: $\operatorname{Card}, <, + , \times,\text{^}$
where $\...
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1
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Analysis I, simpler proof of Tao's construction of the integers [closed]
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
- 236k
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2
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379
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Can singular long models require less than PA?
Say that a long model is an $\mathfrak{A}\models\mathsf{I\Sigma_1}$ such that $\mathfrak{A}$ has strictly greater cardinality than each of its proper initial segments (in the case $\vert\mathfrak{A}\...
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27
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5
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4k
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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 ...
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5
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1
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483
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Extensions of the Ackermann interpretation to nonstandard theories of arithmetic
In their paper, " On Interpretations of Arithmetic and Set Theory" (Notre Dame Journal of Formal Logic, Vol. 8, No. 4 (2007), pp. 497-510) in section 7, "Fragments of Arithmetic and Set ...
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1
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Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning?
By "formal analogies" between the metamathematics of $\mathsf{ZFC}$/set theory and $\mathsf{PA}$(=Peano Arithmetic)/first order arithmetic, I mean facts such as the following:
We are considering a ...
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Can we extend the projectively extended real line with a single number that stands for division of zero by zero?
If we work within $\hat{\mathbb R} = \mathbb R \cup \{\infty\}$, i.e. one point compactification of the real line.
We extend $<$ relation on $\mathbb R$ to $\hat <$ defined as:
$ x \ \hat{<} \...
- 11.3k
-2
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1
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369
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Is this extension of the projectively extended real line, consistent?
This posting has been Edited. The edited material shall be noted.
The projectively extended real line $\hat {\mathbb R}= \mathbb R \cup \{\infty\}$ is one system which allows division by zero! Yet it ...
- 11.3k
6
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1
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Interpretation of $ZFC^-$ in 2nd order Peano arithmetic
Let $Z_2^-$ be the 2nd order Peano arithmetic without the schema of Countable Choice. It has been known, since 1960s at least, that $ZFC^-$ (without the power set) admits an interpretation in $Z_2$ ...
- 2,167
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1
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140
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Can we always know if an algebraic rule over the reals is preserved over the extended reals or not?
Recall a prior posting titled Is there an effective way to generalize this approach of affinely extending the number line?, and especially the accepted answer given to it. So we are working in $\sf ...
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1
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1
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213
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Is there an effective way to generalize this approach of affinely extending the number line?
The general approach here is a follow up of the approach outlined in a prior posting on extending the projectively extended real line. In particular arithmetic operators break down to ternery ...
- 20.5k
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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
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2
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Can we interpret arithmetic in set theory, with exactly PA as the ZFC provable consequences?
There are many interpretations of arithmetic in set theory. The
Zermelo interpretation, for example, begins with the empty set and applies the singleton operator as successor:
$$0=\{\ \}$$
$$1=\{0\}$$
...
0
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0
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152
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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 ...
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2
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1
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147
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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 ...
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5
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0
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317
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$\Sigma_n$-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 ...
- 63.9k
9
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Is there a definable model of PA whose domain is a proper class and whose complete theory is not definable?
Assume ZFC. Is there a formula of $\mathcal{L}_\in$ (without parameters) defining a model $\mathcal{M}$ of PA whose domain is a proper class but the complete theory of that model is not definable by ...
- 20.5k
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Complete and consistent first-order theories that contain interesting phenomena
Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete.
I think there is some sentimental value in working with a theory ...
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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 ...
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11
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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 ...
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How special is first-order $\mathsf{PA}$?
This is a modified version of a question which was asked and bountied at MSE without success.
Below, "$\mathsf{PA}$" refers to first-order Peano arithmetic.
There are various "...
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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. ...
- 63.9k
6
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1
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Nonstandard models of PA of large cardinal size
It is easy to overlook the fact that the existence of a given large cardinal provides us with a true arithmetical statement that would otherwise be false if the large cardinal notion were not ...
- 63.9k
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Uncountable nonstandard models of PA
Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used ...
- 63.9k
3
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0
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Set theories that are complete modulo finite-order arithmetic
In a previous question, I asked whether there can be effectively axiomatizable set theories (at least as strong as, say, ZF) that are complete modulo first-order arithmetic, to which the answer is no; ...
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3
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Are there first-order statements that second order PA proves that first order PA does not?
Are there first-order statements that second order PA proves that first order PA does not? Is this known one way or the other? Could you share an example? (edit: to clarify, by 'second order PA' I don'...
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Could Kronecker accept a proof of Goodstein's theorem?
A famous result of Goodstein asserts that the Goodstein sequence of integers terminates.
For a precise statement and a short proof, see https://en.wikipedia.org/wiki/Goodstein%27s_theorem.
A well ...
- 82.7k
6
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3
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Set theory inside arithmetics via the Ackermann yoga
Among the basic results of logic which, simple as they are, never fail to intrigue me, is Ackermann's interpretation of ZF-Infinity in PA (see for refs this MO question and here for an excellent ...
- 63.9k
3
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0
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301
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What does second order set theory give us that is new?
There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...
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