Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved. — Charles Darwin
Which means ∞∞ is impossible, squaring with Aristotle's decision to make the distinction potential vs. actual (infinity). — Agent Smith
Can infinity even be considered as possible or impossible? it's a concept. If an axiom calls it to be true/in need then I see no reason to assume otherwise. — john27
I haven't read the original argument made by Aristotle - Wikipedia offers only a rough sketch. It seems as though Aristotle considered real/actual entities as those that had an end; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. The same goes for ∞∞, it, by definition is endless. — Agent Smith
If there were an instrument that meausred infinity, then the actuality would immediately show through. — god must be atheist
d; consider the process of constructing a chair. It begins (wood, nails, glue, etc.) and ends (a chair). If one is unable to complete the task, we have a potential chair and not an actual one. — Agent Smith
If, a big if, there did exist a finite number Nmax that could stand in for, salva veritate, ∞∞, we could prove/disprove all mathematical conjectures via proof by exhaustion (brute search) with the help of existing supercomputers — Agent Smith
"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze
"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze
Posters don't ordinarily think "Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze
In acting virtuously, virtuous action becomes habit — Kuro
Question: What's the Nmax for our universe? — Agent Smith
aleph-1. The the infinite cardinal of the real numbers — ssu
- an unsettled question. — TonesInDeepFreeze
Actually, I think the continuum hypothesis is that from aleph_0 the next is aleph_1. — ssu
What did I say? There's still something for us to understand with infinity. — ssu
card(reals) = aleph_1 is the continuum hypothesis. It is not provable in ZFC. It is thought to be true by some mathematicians and false by other mathematicians - an unsettled question. — TonesInDeepFreeze
aleph-1. The the infinite cardinal of the real numbers — ssu
I think the continuum hypothesis is that from aleph_0 the next is aleph_1. — ssu
aleph_1 is the next aleph after aleph_0
is not the continuum hypothesis. — TonesInDeepFreeze
The cardinality of the set of real numbers is aleph_1
is the continuum hypothesis. — TonesInDeepFreeze
Well, it's about what can be put into one-to-one correspondence with the set of natural numbers and the reductio ad absurdum proof that this cannot be done with the set of reals. Here 'countable' has it's problems, when ordinarily everything that we can map into one-to-one correspondence is countable (a+b=c).As to the other poster, the current question of the continuum hypothesis does not stem from the definition of 'countable'. — TonesInDeepFreeze
aleph-1. The the infinite cardinal of the real numbers — ssu
the continuum hypothesis is that from aleph_0 the next is aleph_1 — ssu
what can be put into one-to-one correspondence with the set of natural numbers and the reductio ad absurdum proof that this cannot be done with the set of reals. — ssu
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