Questions tagged [set-theory]
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Relativism and common sense in ZFC
ZFC is the most well known set theory which is considered by many as the foundation of mathematics but I am confused to understand it intuitively. Most of us have a clear understating of empty set and ...
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Jerry-rigging a set's elementhood parameters
Suppose one has a bunch of possible elements x that satisfy some parameter P. Then suppose one tries to get a set X that holds all satisfiers of P, so all those x. Suppose that X doesn't itself ...
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What if all sets contain themselves? [closed]
If we define (i.e., by definition) a set as an abstract collection of at least one element, and
If we say that by nature (i.e., as an axiom) all sets contain one element that is the set itself at the ...
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Would contingently inhabited sets be necessarily impure?
Allow that ur-elements can count as purely set-theoretic, depending on which ones are introduced (we might say: an ur-element is pure if the only extralogical facts about it feature just the ...
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What did Russell mean when he wrote that the null-class, the class having no members, did not exist?
I am not quite sure I interpret the following sentence correctly in Bertrand Russell's paper on existential import:
and among classes there is just one which does not exist, namely, the class having ...
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Is the response (in the mathematics community) to Wiles' proof of Fermat's Last Theorem, evidence for social constructivism about math?
Wiles' proof initially involved reference to functional equivalents of inaccessible cardinals (here, Grothendieck universes). Rather than take this as evidence for the meaningfulness and usefulness of ...
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Is set theory a part of, or even identical to, mathematical logic?
The philosopher Quine famously said that second-order logic is set theory in sheep's clothing. However, what if it is really the other way around? Is set theory part of mathematical logic, or even the ...
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How is the concept of a topos in mathematics relevant to philosophy?
https://en.wikipedia.org/wiki/Topos
Topoi behave much like the category of sets and possess a notion of
localization; they are a direct generalization of point-set topology.
My understanding is that ...
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Is category theory an example of foundherentism?
After reading this essay about the history of type theory, I have refined my assessment of the set- vs. type-theory question in two ways. More similarly to what I was thinking before, I still ground ...
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Is there a difference between "is an intensional element of" and "is an extensional element of"?
There is a version of set theory according to which there are two flavors (types? categories?) of elementhood relation, and if it's ultimately coherent, it does offer a solution to Russell's paradox (...
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Is the multiverse standpoint in set theory "ideologically committed" to plural quantification over universes/axioms?
One of the ways in which Hamkins expresses the multiverse standpoint is as the assertion that there is no "absolute background concept of sets or even ordinals." He spells out examples of ...
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Is the set of all good things incommensurable with the sets we use the natural/related numbers for?
Suppose that there is a set of all good things, and that it is well-founded. Then it would not be an element of itself, i.e. would not be a good thing. Maybe it would be hypergood, but maybe it would ...
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Could we use the foundation axiom to generate counterexamples to almost any substantial axiom?
Here's the argument scheme I have in mind ("F" refers to a substantial/positive property/description; negative qualifiers like "inaccessible" do not sustain this scheme correctly):
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Filling in the gaps in an erotetic argument for pluralism about the Continuum Hypothesis
Syntax assumption/stipulation. I have decided to work with an erotetic function that is parenthetical. So I will not start with some proposition A and then have A? as its associated question, but I ...
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ZFC Axioms and First Order Logic
I have never been formally trained in logic and philosophy. I became increasingly interested in the foundation of mathematics after I graduated from university.
Recently, I've been self-studying ZFC ...
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Is the powerset question really an (indefinitely expansive) series of questions?
At the "end of the day," it has turned out that:
If we deny the powerset axiom, then the expression "the powerset of the zeroth aleph" refers to nothing, in which case there is an ...
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What is the definition of a set of "purely mathematical" objects?
I was originally planning to post this on Math Stack Exchange, but I decided this was a better stack exchange. According to the axioms of set theory, there exists a set containing just an apple, and ...
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Can 'collections' be 'objects'?
Most things we call 'objects' are generally made up of other 'objects' can we consider a collection, such as a physical collection of objects as an 'object' itself?
If we have a 'collection' or an ...
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Physical vs abstract collections
In mathematics we deal with 'sets' they are abstract as the objects in them are abstract, they have no tempo-spatial location. How about standard 'collections' we would encounter in real life, if I ...
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Do set theories have inconsistency strengths, on top of consistency strengths?
Caveat: this question is fairly technical in nature, and I have reason to believe it would be more fitting for the MathOverflow, especially in terms of potentially informative responses (there are ...
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Discerning between a number 'x' as a Natural or Real number
The usual way of teaching is to explain the numbers that are element of the reals and naturals as being the same, this was a perfectly valid way of understanding for me, why do some consider '2' as an ...
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What is the benefit of distinguishing elements of a set instead of subsets of a set?
What is the purpose of "is element of" relation instead of just using "is a subset of" relation for everything?
For example, instead of saying, "Set A has a subset B, and B ...
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If "erotetic model theory" makes sense, what is its relationship to the downward Löwenheim–Skolem theorem?
First, a point-of-departure introduction to/outline of model theory:
An interpretation of an axiomatic system is the assignment of meanings or values to a given axiomatic system such that all ...
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Can the "doctrine of degrees of existence" be used to support the well-ordering lemma apart from the axiom of choice?
I was pleasantly surprised to read (in a Wikipedia article) that:
In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering ...
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Can a collection be a single 'thing'?
Can a collection be any single object in itself, for example Collection A is 'one collection' or set A is 'one set' or even one 'mathematical object' if viewed as a whole? For example a set containing ...
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ZFA vs. ZFC vs. ~Con(ZFA)
Among other places, I've seen it here said that ZFC and ZFA (cofoundational instead of strictly well-foundational) are equiconsistent. I feel like I should know the answer to my question, but if ZFA ...
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Why is a set with one element distinct from the element itself?
Why do we consider a set which is treated for all intents and purposes as a 'collection' with one element as being different from the element itself? In this 'collection' there is one element, and ...
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Constructivism and the effects of constructing definitions on morality
My friend states that 'morality is subjective' since one can construct definitions based on arbitrary intensions and extensions of a well-defined set viz. 'My own morally not bad actions' from the ...
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Constructing natural numbers from nothing
I found that many of us (mathematicians) try to construct natural numbers defined from the intuitive concept 'size of the set'. They take ϕ, the empty set, as the starting point, then define and ...
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If the Continuum's cardinality could physically change because of a physical manifestation of forcing
Suppose that the Continuum has a specific cardinality assigned to it when it exists "in" a physical universe. (This, or a similar-sounding supposition, occurs frequently enough in the ...
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Transconsistency and Diaconescu's theorem
Let T be some set theory. Now there is at least one constructivist T such that T(AOC) ⊨ LEM, that is, via the axiom of choice in T, the law of the excluded middle becomes a theorem (Diaconescu's). So ...
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Is there a notion analogous to “the inductive conception of set” in modal logic?
In many introductory treatments of modal logic, one defines a Kripke model with respect to some domain D. In variable domain semantics, each world in the Kripke model is assigned a different subset of ...
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Can set theory be non-extensional?
Here is Juliet Floyd stating "Wittgenstein's non-extensionalism, like Russell's in Principia, precluded development of an extensional theory of the infinite (set theory). https://youtu.be/...
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Can something be both a type and a token?
I wonder, can something be both a type and a token (in reference to the type-token distinction in philosophy)? For example, an individual dog is a token of the type of dog, but the type of dog is ...
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Intuition behind existence of "function sets"
The usual axioms ensure the existence of certain sets that serve as functions. For example (which is chosen arbitrarily) the function f which maps real values of x to x^2+2 can be represented by the ...
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Cardinality of the logical space according to David Lewis
I just read an extract of David Lewis's Counterfactuals and he claims there in a footnote on page 90 that there are at least beth_2 possible worlds. He also claims in the very same footnote that "...
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Are implementations to semiotics what proofs are to syntax, and models to semantics?
I see this term implementation and its family used here and there in writings on set theory. There are implementations of natural numbers, of ordered pairs, of functions, of "mathematics in ...
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Set theory with full comprehension
A few years ago I was reading an entry in the Stanford Encyclopedia of Philosophy related to set theory and I stumbled across a statement along these lines:
There is a set theory where full ...
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How to write that two specific sets are the same set in first-order naive set theory?
Consider the following true statement
(1) The set whose only members are the prime numbers between 6 and 12 is the same as the set whose only members are the solutions to the equation x^2-18x+77=0.
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Erotetic quantifiers?
I can't find it, but I have a distinct memory of there being an SEP article that says something like as if the ordinals were there before the sets in relation to a question about whether ordinals (or ...
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Can Physical and Mathematical objects exist in the same set?
I'm fairly comfortable dealing with sets of purely Mathematical objects, when I remembered something I heard a long time ago that sets can have elements of any type. Is it possible to define a set ...
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Mathematical "forms" as a relation of varying arity
This might be more a MathSE question, but on the other hand, it would involve a peculiar reimagining of the relation between set theory and type theory, so I'll try it out here.
OK, so earlier I ...
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Set theory vs. type theory vs. category theory
IIRC, in the univalent-foundations program (per Voevodsky), category theory is represented as a possible sort of evolution or new wave of type theory. Maybe my memory is off, but anyway, in nlab they ...
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Rehabilitating Zermelo's description of the universe of sets?
I was reading various essays about Cantor's doctrine of absolute infinity, and it came up again that Zermelo's doctrine, by contrast, was of V as an "unfinished totality." Initially, this ...
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What are sufficient grounds for establishing a theory?
This question delves into the definition of a theory, but somewhat into the grounds of Set Theory as well. I was wondering on what grounds is theory established and accepted. To what degree do the ...
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Can Mathematics be a tool to analyse immaterial existences
Doubt: Can we say that Mathematical thoughts (arguments) can be independent of physical (Time-space continuum or material) world since it is an abstract science. In other words can Mathematics be a ...
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Justification values
The concept of truth values is sometimes expressed in terms of "truth as an object vs. truth as a property." My in-a-slogan understanding of this alternative is "sentences being ...
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Forcing and justification
In "The set-theoretic multiverse," Hamkins talks about forcing giving us "glimpses" of other set-theoretic universes. He states his position as a Platonistic one, i.e. these "...
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How do we arrive at stronger theories in mathematics/logic?
A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us ...
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Does the forcing phenomenon prove some sort of set-theoretic multiverse?
It seems that, "A can be forced to equal B," allows, "A is possibly equal to B." In possible-worlds lingo, this gets us, "There is a possible world where A = B." Since ...
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