Questions tagged [set-theory]
Use for questions about sets, functions on sets, cardinality, set-theoretic axioms, set-theoretic paradoxes, philosophical interpretations of set theory, etc.
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Why are pure powers of the empty set insufficient as a definition for ordinals?
I recently discovered a philosophical term that gives expression to a paradigm that had been circling in my head. G. E. Moore discussed the “paradox of analysis”, which is similar to what I think of ...
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On the Peano Axioms (Set Theory) [duplicate]
According to Wolfram Math World, the Peano Axioms are as follows:
0 is a number
If x is a number, then the successor of x is a number
0 is not the successor of a number
Two numbers of which the ...
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Is infinity a concept or a word empty of meaning?
I'm wondering if infinity is a concept.
We know from experience that there are things for which one cannot reach the end. A long way through the space is an example. One cannot reach the end of the ...
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How do Quinian ur-elements compare to Lawverian abstract sets?
Basically, although ZFC is typically the norm in modern mathematical practice (I am told), there are also well-known issues that suggest it may have its own expiration date.
I did not see the proof, ...
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Issues with cardinality [closed]
How many distinct possible axioms are expressible in a first-order language? My hypothesis is aleph null, countably infinite.
How many possible (finite) combinations of such axioms are there? My guess ...
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Skolem’s paradox
https://arxiv.org/abs/math/0509246
Anomaly #3: Skolem’s paradox.
According to the Löwenheim-Skolem theorem, any infinite model of a countable family of axioms has a countable submodel.
This can be ...
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Why is hypercomputation contested?
Clearly, I do not have a solid grasp on a number of the following topics, and I would like to. I’ll try to explain my reasoning clearly, so anyone could point out any of my misunderstandings.
The ...
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Are identical sets distinct objects?
I have two sets, A and B, both containing the elements 1, 2, and 3. Since they are identical, does this mean they are actually the same single set, just represented twice?
Thinking of sets as boxes, ...
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Why is list of names no more capable of expressing a proposition?
From the Open Logic Project book 2.2, Philosophical reflections (Set theory):
Third: when we “identify” relations with sets, we said that we would allow ourselves to write Rxy for ⟨x, y⟩ ∈ R. This is ...
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Does the empty set exist? [closed]
Does the empty set exist?
I don't think it exists
because
If there are no constituent elements of an object, it can be said that the object does not exist.
Can't this be the reason why the empty set ...
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Would "to avoid the class/set distinction" be, or not be, an ad hoc reason to propose a couniversal set?
Once upon a time, von Neumann proposed the axiom of limitation-of-size, which says that any class "too large to be a set" is then a "proper class," meaning that there is a ...
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First order logic and the cosmological argument
The way I see it, the cosmological argument, if one takes into consideration only what has been observed in the universe, goes something like this:
For everything in the universe, if it has a ...
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The set-or-class of things that don't exist
If it could be determinate, how many things don't exist, i.e. if there could be a set of nonexistent things, would the existence of other things follow "mechanically"?
If it's not ...
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Why is equality not considered the fundamental unit / principle in nature?
(I don't know how to ask this other than by laying out my worldview. Needless to say, this is here to be dissected and disemboweled. I realize the broad sweeps will irk people. I hope that by ...
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Problems with saying that our universe is physically closed (reformulating Kant's antinomies)
Initial caveat: some misapprehension seems to have arisen over my reference to physical sets. But in this, I am trying to follow the language of modern topology, which seems to be applied everywhere ...
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Amorphous sets and vagueness
I'm reading a detailed study of amorphous sets and this caught my eye:
With respect to "epistemicism" about sorites problems, is there some way to correlate the possible (if unidentifiable?)...
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I'm so confused, why doesn't the foundation axiom allow us to derive ℘(ℵ₀) ≠ ℘(ℵ₁), or worse, why doesn't that axiom show that ZFC is inconsistent?
℘(ℵ₀) ≠ ℘(ℵ₁) is not provable in ZFC (this unprovability is an instance of Easton's theorem). I don't know why my mind decided to get hung up on this today, but I'm tired and this brain bee is buzzing ...
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What does a set of pencils contain when we know that pencils are not physically present in the set?
I have some pencils, markers and a mobile phone on my desk. I consider the pencils and think of them as forming a set to which they belong. This set is now a thing existing in its own right.
These ...
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How many variations on graph-theoretic/related parameters indicate alternatives to foundationalism/coherentism/infinitism?
The set theory I'm trying to work in right now is geared towards applying an "axiom of multifoundation" whose local maximum representation is:
The interpretation of the elementhood glyphs ...
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Where does the canon event theory of identity formation come from?
There is an idea in the new Spider-man movie Spider-Man: Across the Spider-Verse, where spider-men through different dimensions have to deal with inter-dimensional problems. In it, one critical part ...
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Peirce cuts (mirrored) + demi-negation = demisets?
[Note: I found one essay, about Aristotle, that used the word "demiset," although at a glance it seemed like they might've been substituting this terminology for a counterpart to the subset/...
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Can the concept of antisets be used for a neo-mechanist causal set theory?
Background information:
"Causal Approaches to Scientific Explanation," sec. 1. My takeaway here is looking at individual existential quantifications, i.e. quantifying over individual causal ...
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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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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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2
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What exactly is the relationship between first-order logic and the axioms of ZFC? Which one is more fundamental?
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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0
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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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3
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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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11
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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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