# Tagged Questions

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### General question: what happens if we replace the regularity stipulation in GCH with other conditions?

I went to bed last night pondering the following. We can formulate both a weak and a strong generalized continuum hypothesis. GCH0. If $\kappa$ is an infinite cardinal number, then $2^\kappa$ is ...
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### Book/Books leading up to the the axiom of choice?

I am familiar with the axioms of ZF set theory and some basic uses of them to completely formally construct more complex objects such as natural numbers etc. However I have pretty much no background ...
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### Recommended books/articles for learning set theory

What is the recommended reading for thoroughly learning set theory? I'm currently studying Kunen's book [1]. But what then, and in what order? One needs to learn large cardinals, inner models and ...
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### Subject-level guide for Princeton Companion to Math?

I have the Princeton Companion to Mathematics, which I'm enjoying overall. However, right now it's a lot more useful to me for expanding on topics I'm already somewhat familiar and less useful for ...
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### The regularity of successor cardinal

I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as ...
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### Reference Request Scott's Trick

Does anyone know of a reference for Scott's Trick. I can't find it in Set Theory-Jech?
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### Founding Arithmetic on geometry

In the past I found some fleeting references that some (Frege in his later years being one of them) tried to found arithmetic not on set-theory and logic but on geometry and logic. Unfortunedly Frege ...
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### Cantor-Schröder-Bernstein without elements

The Cantor-SchrÃ¶der-Bernstein Theorem for the category of sets has several "analogues" for other categories as well, for example measurable spaces and operator algebras (but not for arbitrary ...
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### Consistency strength of Turing measurability

This is probably well-known to recursion theorists, but as google didn't help me, I'll ask it here. Convention: All sets of reals in the following discussion are assumed to be closed under Turing ...
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### Reference Request for GB-set theory

Could anyone give me a reference for a book which has an introduction to set theory from the GB axioms as opposed to ZFC, everything I read seems to just look at things from ZFC (Jech...) Thanks for ...
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### Set theory aspects of category theory

I have never learnt axiomatic set theory, but have studied it from Munkres's Topology book first chapter. So I do not understand the difference between a class and a set except in some vague sense. ...
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### Has anyone successfully axiomatized the category of finite sets? In such a way that the resulting theory is bi-interpretable with PA.

In studies of ZFC, it is conventional to take Peano arithmetic (hereafter PA) as the metatheory. However, I don't like this convention; I think a better approach would be a metatheory (like ZF-fin ...
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### A general form of Tukey's Lemma

In the book "General Topology" by John Kelley, Tukey's Lemma is stated as: "If a family of sets is of finite character, it has a maximal member". In the end of the section on set theory, there is the ...
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### Introducing ordered pairs in an axiomatic way [duplicate]

Suppose that in $ZFC$ we have introduced ordered pairs not in the usual way as $(a,b) = \{\{a\}, \{a,b\}\}$ but axiomatically, by extending $ZFC$ by adding to $ZFC$ a new binary functional symbol $g$ ...
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### Are there mathematical (combinatorial) objects to represent such set systems?

Background: This is a problem arising from my study on computer science. In its appearance, it involves set systems. A set system $\mathcal{S} = \{S_1, S_2, \ldots, S_m \}$ is a collection of ...
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### Forcing Language

Does anyone know of a reference that explains the concept of forcing by fixing a forcing language that has a (I believe unary) predicate and does not mention the forcing poset? For example, Kunen's ...
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### Are there such things as non-extensional set theories?

I have always assumed that extensionality is a paradigmatic example of a property of mathematical objects (sets) which is essential to those objects--- if your set theory doesn't obey extensionality, ...
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### A few questions about the notion of 'proper class' in set theory

In a previous discussion on this website, Asaf Karagila said something to the effect of: Given a model $(V,\in')$ of ZFC, the classes of that model are (by definition) the subsets of $V$ definable ...
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### Con ZF implies Con ZFC using set sized models

Can we use forcing to construct models of ZFC and ZFC + GCH starting from c.t.m s of ZF? The usual way to obtain the associated relative consistency results (Con ZF implies Con ZFC and Con ZF implies ...
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### Foundations of Forcing

I am currently studying Forcing methods in order to understand some independence results and model's constructions. Now I am interested on formalizing the main notions around forcing such as ...
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### Can we found mathematics without evaluation or membership?

In some sense, composition generalizes evaluation. The trick is, instead of writing $f(x)$ for $x$ an element of the domain of $X,$ we write $f \circ x$ for $x$ a function $1 \rightarrow X$. ...
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### Proving equivalence of a tree-based version of Countable Choice for families of finite sets.

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
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### Dependent choice does not imply “the reals are well-ordered”; citation?

As silly as this sounds, I can't find a proof that the axiom of dependent choice (DC) does not imply that the reals are well-orderable. My memory is that this is a fairly early result in the history ...
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### Dependence of Dedekind's Theorem on AC using Scott's trick

I would like some reference about this fact: the essential dependence on choice of Dedekind's theorem in, e.g., Kelley-Morse set theory using Scott's trick.
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### Books on logic, proof theory and set theory?

I graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have only ...
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### Random reals and Martin-Löf randomness

My questions are about the relationship between the following notions of randomness: A real $r$ is random over the model $M$ if $r\notin B$ for every null Borel set $B$ coded in $M$. A real $r$ is ...
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### Modern reference on logic-set theory-foundation

I'm looking for a modern book on logic-set theory-foundation written as the Bourbaki's set theory. I'm particularly interested in a formal exposition of ZFC axiom with logic-set Grothendieck universe. ...
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### Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?

Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set ...
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### Good resource to learn transfinite induction and/or recursion?

I'm currently reading John H. Conway's On Numbers and Games, but without a good understanding of transfinite induction and/or recursion, progress is very slow. What's a good resource for learning ...
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### Prerequisites for understanding Borel determinacy

I have just learned that Gale-Stewart game is determined for open and closed sets, so naturally I'm interested to understand Borel determinacy which seems to be on a totally different level. What's ...
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### Reference request: set theory of sigma algebras

I am studying Billingsley's Probability and Measure. The section on sigma-fields (Section 2) seems to demand set-theoretic reasoning beyond what I have been exposed to so far in undergraduate algebra ...
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### Hydra game and quantum superposition

Goodstein's theorem is not provable in Peano Arithmetic showed by Kirby and Harrington in 1982 [Wolfram Mathworld]. Any reference of a "quantum" hydra game where a head can remain in a state of ...
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### Introduction to Infinitary Combinatorics

What are some good texts for someone interested in becoming acquainted with the "big ideas" of infinitary combinatorics? If you'd like more specificity, assume the reader has respectable mathematical ...