All Questions
Tagged with mathematical-philosophy set-theory
117 questions
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Compatible and incompatible sets [closed]
Definition of the compatibility relation
I have defined a relation $\mathsf{C}$ for sets that captures (to some extent) the notion of compatibility.
In order to do this, we need an operation $': \...
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35
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8
answers
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Why not adopt the constructibility axiom $V=L$?
Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having ...
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15
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1
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Does every model of ZF-foundation have an extension, with no new well-founded sets, where every set is bijective with a well-founded set?
This question follows up on an issue arising in Peter LeFanu
Lumsdaine's nice question: Does foundation/regularity have any
categorical/structural consequences, in
ZF?
Let me mention first that my ...
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14
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2
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994
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Set-theoretical foundations of Mathematics with only bounded quantifiers
It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
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3
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Sets = structured sets without structure
Motivation
There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...
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Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?
Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"):
Definition 8. A cardinal $\kappa$ ...
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20
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1
answer
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Axiom of Choice versus V=L in opposition to large cardinals
Consider the following two observations:
The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals.
The axiom of Choice is incompatible with ...
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Does this axiomatic system satisfy requirements for founding mathematics?
In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
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22
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4
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Does Zorn's Lemma imply a physical prediction? [duplicate]
A friend of mine joked that Zorn's lemma must be true because it's used in functional analysis, which gives results about PDEs that are then used to make planes, and the planes fly. I'm not super ...
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5
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3
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488
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Counting without one-to-one correspondence? [closed]
Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
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1
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What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]
This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
- 11.3k
1
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2
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436
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What is against having distinct membership relations on sets in the Platonic realm?
This question is in connection with the question that I've asked at:
Where do models of false theories exist?
The answer to that question was that any consistent theory can have its primitives be re-...
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1
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1
answer
719
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Where do models of false theories exist?
I have some difficulties in understanding the [uni]verse platonic view. How are we to understand the existence of a model of a false theory? what is the relationship of this model to the platonic ...
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1
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Is it natural to hold that Ur-elements, small & big sets and proper classes exists? [closed]
The topic of this post was shifted to
https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist
Since it was deemed to be a philosophical ...
- 11.3k
2
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0
answers
325
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The universe and multiverse views of set theory from the perspective of $ZFC^2$
(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms ...
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5
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Why are real-valued measurable cardinals never explicitly mentioned in Gödel's "What is Cantor's Continuum Problem"?
It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large ...
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2
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Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? [closed]
I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...
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50
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4
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Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively ...
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45
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1
answer
3k
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Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published ...
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0
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125
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Can $GPK^{+}_{\infty}$ + $AC_{WF}$ prove "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes?
Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...
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9
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1
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How are material set theory and structural set theory related from the point of view of category theory?
In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman ...
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55
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10
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How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
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6
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1
answer
209
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Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set theory?
As is well known, the following theory is equiconsistent with $PA$:
$ZFC$ with the axiom of infinity replaced by its negation.
Since this theory is equiconsistent with $PA$, it would seem ...
- 6,132
11
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1
answer
1k
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Belief in consistency of extremely large cardinals
One of the most common justifications for the consistency of large cardinals is the development of a coherent inner model theory for many large cardinal axioms. While the strength of this argument can ...
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3
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1
answer
182
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What is the weakest large cardinal property which is equiconsistent to weak compact cardinal?
Accoding to wiki, weak compact cardinal is a very weak property in the large cardinal ladder.
But, like ZFC+CH to ZFC, weak compact has some "useless part", so that even the first Woodin cardinal may ...
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22
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4
answers
4k
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Are proper classes objects?
Many of us presume that mathematics studies objects. In agreement with this, set theorists often say that they study the well founded hereditarily extensional objects generated ex nihilo by the "...
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2
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Why are model theorists free to use GCH and other semi-axioms? [closed]
Looking into the open problem section of the book Model theory by Chang and Keisler, I noticed that many problems assumed semi-axioms like GCH. I talk about 'semi-axioms' because these "axioms" are ...
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1
answer
319
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Tarski-Grothendieck in the cumulative hierarchy
How can the Grothendieck-Tarski axiom seen to be true in the cumulative hierarchy of sets?
What are intuitions that would convince us that this axiom is true?
- 73.2k
0
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1
answer
678
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Why do we try to encode every mathematical object as a set? [closed]
Probably everyone of us has seen set-theoretic encodings of mathematical objects which we wouldn't naturally consider to be sets. May it be the "definition" of a function from $A$ to $B$ as a relation ...
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0
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260
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Is $\mathit{GPK}^{+}_{ \infty}+\mathit{BAFA}$ inconsistent (and why does it matter)?
Consider Olivier Esser’s alternative axiomatic set theory $\mathit{GPK}^{+}_{\infty}$. Esser defines it as follows (this from his paper "Inconsistency of The Axiom of Choice with The Positive Theory $...
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11
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2
answers
1k
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Last Status of Feferman's Conjecture on Indefinite Value of Continuum
The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...
4
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3
answers
915
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Compactness of existential second order logic and definability of certain quantifiers
It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with ...
5
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0
answers
323
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Can the Kunen inconsistency (or the existence of Reinhardt cardinals) be 'properly formulated' in Ackermann set theory?
In their paper "Generalizations of the Kunen Inconsistency" (arXiv:1106.1951v1 [math.LO]10 Jun. 2011), Hamkins, Kirmayer, and Perlmutter write the following:
The first [metamathematical issue--my ...
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-2
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1
answer
281
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Critical points and the Foundation Axiom
(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the Stanford Encyclopedia of Philosophy entry "Non-Wellfounded Set Theory" (...
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12
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3
answers
3k
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Has Dedekind's proof of existence of infinite sets been analyzed by historians?
This pdf by David Joyce notes that in paragraph 66 of his famous essay, Dedekind claims to prove the existence of an infinite set.
The proof exploits the assumption that there exists a set $S$ of all ...
- 6,132
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vote
0
answers
265
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Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried out in positive set theory?
This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems (...
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13
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1
answer
751
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Is there a class of mathematical structures with non-isomorphic natural representations as a standard Borel space?
Background. The field of Borel equivalence relation theory
provides a robust, unifying theory that organizes most of the
classification problems of classical mathematics into a hierarchy,
allowing us ...
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9
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2
answers
2k
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Using the multiverse approach to decide the law of the exluded middle?
Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
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18
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2
answers
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Universe view vs. Multiverse view of Set Theory
Here I refer to Hamkins' slides:
http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf
particularly, to the "Universe view simulated inside Multiverse", p. 22.
My question is: is it very unsound ...
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9
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3
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1k
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The universe of sets, existential quantification in set theory
Yesterday, I posted a question that was received in a different way than I intended it. I would like to ask it again by adding some context.
In ZF one can prove $\not\exists x (\forall y (y\in x)).$ ...
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9
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1
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856
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Taller models of ZFC
This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.
Using forcing techniques, at least the ones I know of, one starts from a ...
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9
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3
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3k
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What does the axiom of replacement mean and why should I believe it?
Here Professor Blass describes the following cumulative hierarchy of sets:
Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), ...
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13
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0
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882
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Arguments against Freiling's argument against Continuum Hypothesis
Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
2
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0
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134
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A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability
Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...
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1
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1
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446
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A question regarding extendible cardinals and a result of M. Magidor
The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called $\...
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2
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580
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A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals
A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there ...
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4
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1
answer
386
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Plausibility argument for a measurable cardinal
The following question is not mathematically precise but perhaps of some philosophical interest.
A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...
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2
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1
answer
880
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Is second-order ZFC categorical with regard to its proper class models
Second-order ZFC offers partial categoricity in the sense that, given any two models, one of them must be isomorphic to an initial segment of the other [1]. However, this leaves questions regarding ...
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2
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The impact of large cardinals in mathematics [closed]
What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...
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2
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Can we define an "empirically generic" real number?
Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
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