0
votes
1answer
40 views

Godel's completeness theorem and formula that states consistency of ZF

Godel's completeness theorem, in original formulation, says that every logically valid statement/formula has finite deduction of a formula. Now then there is Godel's incompleteness theorem. Would this ...
1
vote
0answers
39 views

Are there axiomatizations of first order logic or set theory defined in first order logic or set theory?

There are several axiomatizations for number theory, group theory, and other theories represented in first order logic. Further, these theories are also representable in set theory such as $\sf ZFC$ ...
3
votes
1answer
46 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
1
vote
1answer
52 views

Comprehension and Impredicativity

Wang and McNaughton (Les Systemes Axiomatiques de la Theorie des Ensembles, 1953) discuss briefly the topic of impredicativity in chapter 2 (titled 'Type Theory') of the above mentioned book, but I'm ...
2
votes
1answer
41 views

If a version of GCH holds for Chang's $\kappa$-constructibility, does a version of GCH hold for $L_{\infty}$?

In C.C. Chang's paper "Sets Constructible Using $L_{\kappa \kappa}$" one can "deduce a version of the GCH, theorem VI [(iv)--my comment], assuming that all sets are $\kappa$-constructible." Now ...
1
vote
0answers
77 views

A Question Regarding Consistent Fragments of Naive (Ideal) Set Theory

It is well known that Naive (to some, otherwise known as Ideal) set theory, that is, the set theory generated by the axioms: (EXT) (x)(x $\in$ A iff x $\in$ B) iff A=B (COMP) ($\exists$y)(x)(x ...
6
votes
1answer
78 views

Construction of Ultrafilters

I've been doing a lot of work with ultrapowers and saturation recently. In particular, I am reading chapter 6 of Chang and Keisler as well as Keisler's paper on "Ultraproducts which are not ...
0
votes
2answers
113 views

Is the “Most Important Property a Set S has” Necessary and Sufficient to Define a Paradox-Free Notion of Set?

About a year and a half ago, while I was looking on the Web for papers regarding the Russell paradox, I chanced to find an interesting concept. This concept was contained in what (for want of a ...
6
votes
1answer
174 views

What is semantics of “type”? Do “types” of “type theory” semantically differ from “set” of set theory?

"To be of a (certain type)" is a fundamental relationship for ontology and the computer science "ontologies" are in the core of Semantic Web (which is my interest). But I did not encounter a ...
0
votes
2answers
58 views

Ordinal existence

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$? Could you please suggest me how to even try to solve this?
2
votes
1answer
103 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
5
votes
1answer
68 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
0
votes
1answer
43 views

Examples of non-trivial determiner formulas of trnsitive models of ZFC

Notation: For each $\{\in\}$-formula $\varphi(x_1,\cdots,x_n)$ and each $\in$-model $M$, define: $$\varphi (M)=\{(a_{1},\cdots,a_{n})\in M^{n}~|~\langle M,\in\rangle\models \varphi ...
2
votes
2answers
149 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
6
votes
2answers
247 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
3
votes
1answer
69 views

The order-theoretic structure of the ordinal numbers in non-standard models

I have read the following. Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a ...
3
votes
0answers
83 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
2
votes
1answer
65 views

Absolute confusion! (A question about absolute *sentences*)

I'm seriously confused about absoluteness. A formula in the language of a theory $T$ is absolute for $T$ structures if its truth value is the same in all standard transitive models of $T$ (this may ...
2
votes
0answers
99 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
1
vote
1answer
47 views

Models of ZFC as Ordered Fields

Question: Let $M$ be a set such that $\langle M,\in\rangle\vDash ZFC$. Consider the partial order $\langle M,\subseteq\rangle$. Using order extension principle there is a linear order $\subseteq^{*}$ ...
1
vote
1answer
97 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
votes
2answers
104 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
0
votes
1answer
64 views

Forcing Preservation on Arbitrary Set Theoretic Formulas

Fixing a ground model $M$, a forcing notion $\mathbb{P}$ is called cardinal preserving iff for all $\mathbb{P}$-generic filter $G$ over $M$ we have: $$\forall a\in M~~~(M\models ...
4
votes
0answers
62 views

How to think about iterated ultrapowers?

I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, ...
6
votes
0answers
172 views

Is Kunen's claim about non-equivalent forms of Axiom of Choice, true?

Consider the following forms of the axiom of choice: $AC_1:\forall F\neq \emptyset~~~(\emptyset\notin F~\wedge~\forall x,y\in F~~~(x\neq y\rightarrow x\cap y= \emptyset))\rightarrow \exists C~\forall ...
1
vote
1answer
71 views

Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
1
vote
1answer
37 views

Model for replacement

If $R_{k} $ is a model for replacement, is necesarly k a strong limit cardinal? And if k is a regular cardinal, are then equivalent the sentences: k is a strongly inaccesible cardinal if and only if ...
0
votes
1answer
46 views

About the definible sets $L_\alpha$

Let $\alpha$ be an ordinal number. Is that true that $\alpha$ = $\beth_\alpha $ is equivalent to the statement $|L_\alpha|=|R_\alpha|$, where $L_\alpha $ is the $\alpha$-th stage of the constructible ...
7
votes
1answer
57 views

Every elementary submodel of $H(\aleph_1)$ is transitive

I am trying to solve the first part of Exercise II.17.30 in Kunen's Foundations of Mathematics which asks: Prove that every $A \preccurlyeq H(\aleph_1)$ is transitive. Here we are working over ...
4
votes
1answer
76 views

Ultraproducts of models of ZFC

Let $D$ be a non-principal ultrafilter over $\mathbb{N}$. Let $A_i$ for $i\in\mathbb{N}$ be (countable) models of ZFC such that $A_i\models \mathfrak{c}=\aleph_i$. Then, what is the size of the ...
1
vote
1answer
78 views

Is there a formalisation of set theory where unions can be taken over some classes that are not a priori known to be sets?

Of course, fully unrestricted unions in ZFC will immediately lead to inconsistency, but is there a variation where restrictions on allowed classes are given explicitly at least, in terms of the ...
4
votes
0answers
50 views

Can existence of aleph one be proved without the power set axiom? [duplicate]

Cantor's construction of $\aleph_1$ is to notice that all ordinals constructed after $\omega$ are countable, take the union of all countable ordinals, and then show that this union can not be ...
5
votes
2answers
168 views

About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
1
vote
1answer
80 views

Hall's marriage Theorem and Tychonoff Theorem

I was reading this paper. In particular the second point. He proves the Hall's marriage Theorem for infinite family using the Tychonoff theorem on topological product of compact $T_2$ spaces and the ...
6
votes
1answer
199 views

New Axioms of Infinity

Axiom of Infinity says there is an inductive set (i.e. a set which includes $\emptyset$ and is closed under successor operator). Formally: $Inf:\exists x~(\emptyset\in x~\wedge~\forall y\in ...
2
votes
1answer
83 views

Can the consistency of ZF be proved in MK?

When adding choice to Zermelo–Fraenkel (ZF) set theory, one can go further than ZFC, by first adding classes, then adding global choice, and then proving that the resulting von Neumann–Bernays–Gödel ...
5
votes
1answer
193 views

Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...
3
votes
1answer
109 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
1
vote
3answers
138 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
5
votes
1answer
275 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
3
votes
6answers
689 views

Question about the Continuum Hypothesis

The Continuum Hypothesis hypothesises There is no set whose cardinality is strictly between that of the integers and the real numbers. Clearly this is either true or false - there either exists ...
3
votes
1answer
90 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
2
votes
1answer
71 views

Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
1
vote
2answers
128 views

Existence of an axiom question in relation to $\mathsf{Infinity}$

Original Post This may be a stupid question, but does there axist an axiom $\phi$ that is independent of $\mathsf{ZFC}$, and not equivalent to the axiom of $\mathsf{Infinity}$, such that ...
2
votes
1answer
94 views

Well-ordering the reals: finding a certain model of $\mathsf{ZFC}$

How would one go about constructing a model $\mathfrak{M}$ of $\mathsf{ZFC}$ such that under $\mathfrak{M}$, no formula defining a well-ordering of $\mathbb{R}$ exists? I am certain such models are ...
0
votes
0answers
47 views

The lexicographic order [duplicate]

If it is given ordinals $\alpha$ and $\beta$, the lexicographic order on $\alpha \times \beta$,$\leq_{lex}$ is given by: $(\gamma_0,\delta_0)<_lex(\gamma_1,\delta_1)$ if and only if either ...
2
votes
2answers
126 views

$\mathsf{ZF}$ is not finitely axiomatizable

As we know a first order theory $T$ is finitely axiomatizable if there is a finite set $F\subseteq T$ of axioms such that $F\vdash \sigma$ for every $\sigma \in T$. How we can prove if $\mathsf{ZF}$ ...
4
votes
3answers
108 views

How are models constructed?

As I understand, a model $\langle S,\sim\rangle$ of $\mathsf{ZFC}$ is a set $S$ coupled with some binary operation $\sim$, in which the axioms of $\mathsf{ZFC}$ hold (please correct me if I am wrong). ...
1
vote
1answer
74 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
1
vote
0answers
37 views

How is it possible that the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic? [duplicate]

If I am not wrong, the well-ordering theorem is strictly stronger than the axiom of choice in second-order logic. I am not sure to understand how this is possible. The reason is that second order ...