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1 vote

Why are pure powers of the empty set insufficient as a definition for ordinals?

As Conifold and Michael Carey mentioned in the comments, the Zermelo ordinals do not extend to infinite sets. I want to emphasize this because it's very important to the way ordinal numbers are used. ...
Invariance's user avatar
6 votes

Why are pure powers of the empty set insufficient as a definition for ordinals?

The following sequence is called the “pure” or “irreducible” power sets of the empty set: {}, {{}}, {{{}}}, {{{{}}}}, … Conceptually, this sequence ‘represents’ order, to me. Why is it insufficient, ...
ac15's user avatar
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1 vote

Is infinity a concept or a word empty of meaning?

The concept of inifinity seems perplexing but we must remember that the basic definition is quite simple. It is just anything "without and end". I can construct a set S, which firstly ...
AccidentalTaylorExpansion's user avatar
0 votes

Is infinity a concept or a word empty of meaning?

Perhaps it's best to think of infinity as being foremost an abstraction. Yes, it's something we can't touch and feel. But there are many similar ideas that we find it useful to talk and reason about: ...
Michael Kay's user avatar
0 votes

Is infinity a concept or a word empty of meaning?

Aristotle distiguished between the All and the Infinite. The former is understood just what its conventional meaning states - it is everything. Whilst infinite means not finite. So no matter how much ...
Mozibur Ullah's user avatar
2 votes

Is infinity a concept or a word empty of meaning?

Since your question directly refers to an issue of 'meaning', my answer begins with a reference... Michael Dummett's excellent paper on Intuitionist logic... https://sites.unimi.it/zucchi/NuoviFile/...
21stCenturyParadox's user avatar
1 vote

Is infinity a concept or a word empty of meaning?

The one-to-one correspondence approach in set theory is based on Cantor-Hume principle, which postulates that two sets, whose elements can be set in one-to-one correspondence are equally big. On this ...
Anixx's user avatar
  • 370
0 votes

Is infinity a concept or a word empty of meaning?

Infinite set has various rigorous definitions in mathematical theories. It’s a property of sets, but each infinite set is considered an abstract mathematical object. Classes of infinite sets, such as ...
Davislor's user avatar
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2 votes

Is infinity a concept or a word empty of meaning?

Your comments about it being "obvious" that there are more numbers than even numbers is a bit misguided. The fact that there is a one-two-one correspondence shows that the Cardinality of the ...
Michael Carey's user avatar
-2 votes

Is infinity a concept or a word empty of meaning?

Infinity is a concept. A concept is that which can be indicated but not necessarily observed. To play peek-a-boo with a baby is to experience a tiny moment of absolute joy. Over and over you hide ...
JOHNS WOOD GADGETS's user avatar
0 votes

Is infinity a concept or a word empty of meaning?

All mathematical objects are purely conceptual. Math defines conceptual worlds built on sets of axioms and builds, via logic, more and more complex statements that hold in that conceptual world. Those ...
Harrison Paine's user avatar
10 votes
Accepted

Is infinity a concept or a word empty of meaning?

Mathematics shows that we can make a one to one correspondence [of the] natural numbers with [the] even numbers. This is not right If there is a last number. There are infinite sets that look exactly ...
ac15's user avatar
  • 1,422
3 votes

Is infinity a concept or a word empty of meaning?

I am not sure if this is a proper answer but here goes nothing. I believe, Infinity is a concept, it is not a number but A CONCEPT, a concept of INCREMENT at that (the manipulation of increment beyond ...
How why e's user avatar
  • 1,201
6 votes

Is infinity a concept or a word empty of meaning?

Literally, infinity is both a concept and a word- that should be clear to you from the fact that you have been thinking about the concept and have typed the word more than once in your post. Infinity ...
Marco Ocram's user avatar
  • 21.7k
11 votes

Is infinity a concept or a word empty of meaning?

Your comment seems to be the nub of your problem: I cannot think about infinity because my finite meaning cannot grasp even conceptually non finite objects, despite that there are perhaps infinite ...
Rushi's user avatar
  • 2,801
3 votes

Is infinity a concept or a word empty of meaning?

Infinity is a concept which has been clarified by mathematics. Definition 1: Two sets have the same number of elements (i.e. the same cardinality), if there is a bijective map between both sets. As a ...
Jo Wehler's user avatar
  • 31.7k
2 votes

Issues with cardinality

Is this why we need the axiom of replacement? I haven’t yet grasped what it does. [...] It says 'the universe is closed under definable functions', in that it guarantees that for a (first-order) ...
ac15's user avatar
  • 1,422
1 vote

Issues with cardinality

I'll respond to (4) at length, because the gaps/errors in the presentation are perhaps more subtle than with respect to the other questions. Firstly: There does not appear to be a “biggest set”, due ...
Kristian Berry's user avatar
2 votes
Accepted

Why is hypercomputation contested?

Basically, one would like to know where in the cumulative hierarchy of sets (at least in ZFC) we first see a set-theoretic description of something beyond a Turing machine. Maybe this is provably ...
Kevin's user avatar
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