... it is impossible that an infinite set has a cardinality. — Metaphysician Undercover
I am ready to accept it, as soon as all inconsistencies and contradictions are removed. — Metaphysician Undercover
Those who are "in denial" will always refuse to face the fact that their "well established truths" are actually falsities. — Metaphysician Undercover
Are you ready to address my post now, and show me how an infinite set has a cardinality? — Metaphysician Undercover
As explained here, "Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets."
So, if there is a bijection between elements of two sets then these two sets have the same (and so have a) cardinality. In the case of finite sets, the cardinality is equal to the finite number of elements in the set. In the case of infinite sets, the cardinality is a stipulated aleph number, which in the case of the natural numbers is ℵ0.
Which is why your question doesn't make sense. Infinite sets have a cardinality by stipulation. As aletheist explains about, cardinality is defined in such a way that infinite sets have one. — Michael
And following from this, a countable set is defined as a set that has the same cardinality as the set of natural numbers. — Michael
OK, so there is a different definition of cardinality for finite sets then there is for infinite sets, the former relates to bijection, the latter to stipulation. Which one is used in the determination of "countable" — Metaphysician Undercover
The "cardinality" of the infinite set is not the same as the "cardinality" of the finite set. If "countable" is defined relative to the cardinality of the infinite set, then by definition, the finite sets are not countable. — Metaphysician Undercover
Any one can see that the "aleph numbers" are not the same as the "natural numbers", therefore it is not the same definition. — Metaphysician Undercover
No, it's the same definition and both relate to bijection. It's just that the cardinal numbers used are different. In the case of finite sets we use natural numbers and in the case of infinite sets we use aleph numbers. — Michael
So the set of natural numbers is "countable" according to the value of the aleph numbers? I assume that the aleph numbers negate the infinity of the natural numbers, by stipulating a limit to that infinity. What kind of infinity are we left with then, if it is a limited infinity? Isn't this a contradictory infinity? Of what value are the aleph numbers, if they do nothing other than produce a contradictory infinity? — Metaphysician Undercover
A floozable set is a set with the same cardinality as some subset of the set of natural numbers. — Michael
You're getting too tied up with the term "countable". As suggested above, just consider the term "floozable". A floozable set is a set with the same cardinality as some subset of the set of natural numbers. — Michael
Furthermore, I believe that if we adhere to this separation, mathematics will be rendered useless, because we will not be able to use it to actually count or measure anything. If the principles of mathematics have no relation to what it means to actually count, or measure something, then what good are they? — Metaphysician Undercover
This right here is precisely the reason why we have been at such loggerheads throughout this discussion (and others). As I keep saying over and over, mathematics is the science of drawing necessary conclusions about ideal states of affairs; it does not pertain to anything actual, except to the extent that we use it - with varying degrees of accuracy and success - to model the actual. — aletheist
You have an idiosyncratic metaphysical prejudice that requires something to be actually possible in order to be considered possible in any sense. Again, your worldview is too small; there is much more to mathematics than merely counting and measuring things, and the value of pure mathematics - like that of pure science - is not limited to its practical usefulness. Do not block the way of inquiry! — aletheist
Without moving to exclude those ideals which are impossible, we have no way to increase accuracy and success. — Metaphysician Undercover
If you want a metaphysics which allows that anything is possible, you go right ahead and adopt that metaphysics, but it's not for me. — Metaphysician Undercover
I would prefer to exclude things which at first glance may appear to be possible, but which are later shown not to be possible, as impossible ... the way to succeed in inquiry is to narrow possibilities, by eliminating unjustified possibilities. — Metaphysician Undercover
Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. — aletheist
I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it. — tom
That does not follow. It must be proved. That's Cantor's theorem. — fishfry
It is a mistake to treat accuracy and success in the actual world as the only legitimate objectives of inquiry. For one thing, it is inconsistent with what most people mean when they talk about "ideals." — aletheist
It is a mistake to confuse mathematics with metaphysics. Many things are possible within mathematics that are not actually possible. I also happen to believe that there are real possibilities that are not actually possible, but that is not at all the same thing as allowing that anything is possible. — aletheist
If it is a theorem that has been proved, then it follows, does it not? What am I missing? — aletheist
For any value of N whatsoever, 2^N > N. Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. — aletheist
I may have been responding to you, but I thought it obvious I was not referring to you! — tom
We know that the present is real because of the radical difference between future and past — Metaphysician Undercover
The is no paradox if one a treats time and space as indivisible - which is clearly the case. — Rich
I always get a little uppity when people try to dismiss Zeno's paradoxes with the fact that an infinite series can have a sum. It misses the point entirely. — Voyeur
Moving back toward the original question of this thread, I'm eager to introduce the notion of Supertasks to the conversation. A great summary with examples of Supertasks can be found here — Voyeur
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