I think actual infinity has no useful applications — Devans99
The problems can up start once you try to do anything logical with it. It's not a logical concept so it leads to paradoxes. Particularly if you insist (like set theory does) that infinity is measurable and has a size then it leads to paradoxes. — Devans99
On the number line between zero and one: are there more rational than real numbers? Or the other way round? To know, you have to have a way of quantifying both. How do you go about these? — tim wood
This is a nonsensical definition: for instance, it claims the even numbers are the same size as the natural numbers (as there is a one-to-one correspondence between the two). But the even numbers are a proper subset of the natural numbers. If either had a size, the size of the natural numbers must be greater than the size of the even numbers. — Devans99
If you use a reasonable definition of infinity: ‘A number bigger than any other number’ then it is clear that there could only be one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity — Devans99
Well, the problem is to give a definition of "size" for sets that you cannot count — Mephist
The conclusion could even be that "the measure of the set of all sets does not exist" this is an assumption too.1.Let S be the set of all sets, then |S| < |2^S|
2. But 2^S is a subset of S, because every set in 2^S is in S.
3. Therefore |S|>=|2^S|
4. A contradiction, therefore the set of all sets does not exist.
What is wrong with this ‘proof’? — Devans99
Naught and c; whether c equals 1 is an open question. And nothing else? That's just plain wrong. — tim wood
↪Mephist Sorry, using 2^S to denote the power set of S. The proof I gave is meant to show that the set of all sets does not exist. I maintain that it is the cardinality of the set of all sets that does not exist. — Devans99
It is the assumption that infinite sets are measurable that invalidates naive set theory. ZF set theory is patchwork of hacks that tries to cover all the the holes and fails - the solution is to acknowledge infinite sets do not have a cardinality / size. — Devans99
↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:
∞ + 1 = ∞
This assertion says in english:
’There exists something that when changed, does not change’
Thats a straight contradiction. — Devans99
Well.. I don't like it too, but nobody has shown that is inconsistent yet, and it's used since a very long time. So, I would guess that it's not inconsistent! — Mephist
Sorry, I don't know transfinite aritmetics.. :sad: But if you have some good links to documents that explain what is it I would be interested! — Mephist
But isn't an infinite set bigger than any finite set? Doesn't that imply size, even if obviously you cannot measure it like a finite set?- An infinite set does not have a cardinality property: cardinality or size implies the ability to measure something. Infinity is by definition unmeasurable so infinite sets have no cardinality/size property — Devans99
How do you come to this conclusion? Give a proof that this isn't so. Because the proof of this being so is for me quite understandable (if I remember it correctly): you can well order the rationals, hence there is a bijection between the rationals and the natural numbers.I would not use bijection to arrive a conclusion about a sets size; bijection claims that there are the same number of rationals as naturals so it is clearly wrong (naturals are a proper subset of the rationals so bijection is giving a wrong results). — Devans99
Yeah maybe, you cannot just say that set theory is wrong. It would be similar that I would simply declare quantum physics wrong in physics. You would actually need to give a proof of it in mathematics.To all of us: discussion with Devans99 is a waste of time. — tim wood
But isn't an infinite set bigger than any finite set? Doesn't that imply size, even if obviously you cannot measure it like a finite set? — ssu
Yeah, you cannot just say that set theory is wrong. You would actually need to give a proof of it in mathematics. — ssu
Ok, let's think about what you said here.An infinite set is unmeasurably bigger than a finite set. An infinite set therefore has no size. — Devans99
So, could it be then there would be Absolute Infinity?I'll define infinity as ‘A number bigger than any other number’ then it is clear that there can be only one such number - if there was a second infinity then both would have to be larger than the other - a contradiction - so there can be only one infinity. — Devans99
Meaning that they do have a size, but the size is obviously unmeasurable — ssu
So, could it be then there would be Absolute Infinity? — ssu
Aha!If the size is unmeasurable then it is not a size. Size has to be an integer. — Devans99
In mathematical terms, "size is a concept abstracted from the process of measuring by comparing a longer to a shorter"
↪Mephist How do you justify transfinite arithmetic? The rules of transfinite arithmetic assert that:
∞ + 1 = ∞
This assertion says in english:
’There exists something that when changed, does not change’
Thats a straight contradiction. — Devans99
orry, I wanted to write "finitary", in the sense of "recursively enumerable" (of course not finite, if you can build natural numbers with sets) — Mephist
However, this idea is not mine: (https://www.youtube.com/watch?v=UvDeVqzcw4k) see at about min. 8:23 — Mephist
very non contradictory axiomatic theory based on first order logic has a finite (non-standard) model — Mephist
So, it's impossible to deduce that ∞ - ∞ = 0, and it's not allowed to use the expression ∞ - ∞ as if it was a definite cardinal number.
To derive 1 = 0 from ∞ + 1 = ∞ you should subtract ∞ from both sides of the equation, but
"1 + ∞ = ∞" does not imply "(1 + ∞) - ∞ = ∞ - ∞" because ∞ has not an unique inverse.
So, no contradiction! :-) — Mephist
If he said there are finite models I'm sure there are! — fishfry
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