I mean both “size” and “cardinality” to mean “number of elements”, just as ALL of the definitions state. You have not come up with anything to disagree with that. I can’t think of how to use “size” to mean anything else. So I am going to proceed expecting you to understand that both “size” and “cardinality” mean “number of elements”, which is the only concern here.
If we have a set of items with size/cardinality/“number of elements” of infA (as previously specified) off to our right and another set of items of a different nature off to our left and also of size infA, then the total size of the combined set is 2 * infA.
Phyllo, do you still disagree with the idea that infinite comes in sizes? A) Yes, I disagree - There is only one size of infinite B) No, I agree - There are different sizes related to infinite
or Forfeit
The argument that “…” must always refer to only one size of infinite else be a useless symbol is a bit silly. That is like saying that “apple” must always refer to “McIntosh apple”, else we would never know what we meant by “apple”. Or that “dog” must always refer to German Shepard, else we would never know what “dog” meant.
And you seem to be willing to disagree with mathematicians and Scientific America on the issue of multiple sizes of infinite, so that might end your participation in this debate unless you can come up with some reason why the rest of us should also disagree with them.
You are mixing up ‘units’ and ‘quantities’. If I have a fruit stand then I may have 10 apples and 5 oranges. That means that I have 15 fruits. If I sell 5 fruits then that can mean any combination of apples and oranges which adds up to 5. The basic unit of my mathematics is ‘fruits’.
But what you are doing is redefining what a quantity means for the “…” symbol. You are saying that 3 apples sometimes means 3.5 apples.
That is exactly what YOU are doing.
YOU are disagreeing with the mathematicians in this case.
And also you are saying that if you ADD something (not nothing/“0”) to X, you still have exactly what you had before:
X + Y = X.
3 + 0.5 = 3
This is exactly the case in which your sloppy insistence on using ‘number of elements’ in place of ‘cardinality’ gets you into trouble. ‘Number of elements’ works for when the size of a set is finite. It doesn’t work when it’s infinite. Infinity isn’t a number, so it can’t be the number of elements in a set.
All countable infinities have the same cardinality, in that sense they are the same size. InfA + InfA = InfA. c.f. Aleph naught:
InfA+InfA is countably infinite, so it has the same cardinality as InfA, i.e. Aleph Naught.
Cardinalities are a bit of a red herring here. A decimal expression by definition is a function from the natural numbers N = {1, 2, 3, 4, …} to the set of digits {0,1,2,3,4,5,6,7,8,9}. For example .333… stands for the function that always returns 3. f(1) = 3, f(2) = 3, f(3) = 3, etc.
Likewise the digits to the right of the decimal point of pi = 3.14159… are given by the function f(1) = 1, f(2) = 4, f(3) = 1, etc. If you want the zillionth digit you just look up the value of f(zillion). There’s no end, but there are exactly a countable infinity of digits, one for each natural number.
The decimal representation of a real number consists of exactly a countable set of digits. Uncountable cardinalities are interesting for sure; but not relevant in this context.
When said, ‘all countable infinities have the same size’ it is a matter of going beyond the limits of language. Infinities cannot have either cardinality nor size, because they lack one thing in common: measurement of size.
It’s odd to signify f+1,…+2…, ->, because infinity includes all succeeding numbers. Here again, nomenclature defies it’s own value. This defiance contributes to it’s negative value, it’s contradiction.
Infinity is only helpful as an aim, or derivative within an infinite function. This is why, the mathematical analysis of the question does 1=.n(9)0 where n-<-f , always is at least minimum 90% of total size, so if size can be measured, the infinite sum is never under that, and the maximum, never over 1.0. The differentiation of the sum of the remainder 10% can never have a sum less then that, and that sum is within that particular infinity. The point is, no matter how many operations result in an infinity minus 1/(n)-f operation, the sum will always be (f)-1 sum.
To speak of 1f as an infinite boundary causes the contradiction, for infinitie is not bounded, except , as has been suggested by James, paraphrasing Cantor.
Therefore notation, whether it be mathematical or semantic, can never resolve this basic contradiction.
Simply said, can an infinite function ever become one with it’s Cardinal boundary? No. The real boundary is quite a different matter, because it has to contend with real applications.
Maybe its not your intention, maybe it’s just how you are.
I mean I read one of your dream diaries and half the words were talking about french idioms that I have never heard before in my life…probably weren’t even real idioms you probably just made them up… I am willing to bet not one person in this forums knows what you are talking about half the time…surprise me and prove me wrong? I mean have you noticed how half of your posts seem to just linger at the end of a thread, without any comments…people are too busy trying to analyze it and think of what to say…
You make a good point in this post. I find cardinalities interesting, so I’m a little over-eager to talk about them. But I concede that they do not apply as I suggested.
But don’t the hyperreals raise a similar issue? Since infinity is a number in the hyperreal system, as you describe decimal places as a function, f(infinity) is proper, so could it be that f(infinity)=3 for .333… and f(infinity)=0 for 10(.333…). I don’t know if that’s a matter of cardinality; I get the impression that cardinality works differently for hyperreals.
Ultimate, The idea of spheres as ideal objects is not new. The fact that no line is truly straight led some thinkers back in the beginning of modern philosophy to think of all curveturea being an arc of some sphere. I met Your objections with this idea, which pretained to the ideas under diacussions of limits, by bringing in another seemingly infinite sequence of pi, which also seems endless. Some guy in Japan thought that by veru large repetition of the functional function pi, he could finally arrive at a finite number. However he did not and hence, the idea of a limitless pi goes on appearently
If pi is limitless, the analogy to this forum’s argument is appropriate
I brought in this special example to your objection because the sphere was the ideal mathemati al object entertained by philosophers and mathematicians at that time, and some, still harbor these thoughths enimating from Leibnitz through Kant, and even into modern thought, as relevant.
You have a number of logic problems with that response.
Consistent definition of “add” or “plus”/“+”
Consistent definition of “cardinality” (although irrelevant)
The use of bijection does not confirm your conclusion, but rather refutes/disproves it.
1)
Now is the time to remember the very first question that I asked concerning logical contradiction. You agreed that our arguments must not allow logical contradiction (forgiving poetry). So I now ask for you to examine what the word and concept of the verb “add” means.
To add something A to something B is to change that something B. If the something B didn’t change in any way, then nothing was added. To accomplish adding, there must be something additional. If there is nothing additional, there was no adding. It is a contradiction to say that you added if there was no change.
When you say “infA + infA = infA”, you are contradicting what the concept of “plus”/“+” or “addition” means. In this case, it doesn’t matter whether you call it the same “cardinality”, because no addition to the size actually took place if the size didn’t change.
Thus “infA + infA = infA” is an oxymoron (logical contradiction).
In this debate, contradictions are not allowed (even if allowed elsewhere).
2)
You have agreed that something is the same cardinality if and only if there is a one-to-one, “bijection” relationship between the two sets (as per common stipulation of equinumerosity). Yet infA cannot have a bijection relation with infA1+infA2.
infA can count infA1 in a one-to-one fashion. And separately infA can count infA2. But to count both infA1 and infA2 together, infA must complete the count of infA1 before continuing to count infA2. If infA is said to have completed the count of infA1, then, because they have equinumerosity, the infA must be completely used up, leaving nothing with which to count infA2.
Thus there is no bijection/equinumerosity relation between infA and infA1+infA2 and therefore they are NOT the same “cardinality”.
3)
As outlined in (2), the bijection stipulation leads to the proof that indeed, there is no bijection relation between infA and infA1+infA2 and thus it is proven that: infA+infA ≠ infA
Yes, and none of those terms express the difference between 1.0 and .999…
The difference between 1.0 and .999… is 0.000… with a theoretical 1 at the end. Except, there is no such thing. By your own argument, you can’t stick another number at the end of an ellipsis. if there was such a thing as “0.000…1” or however you want to denote it, it would have a number of qualities demonstrating that it isn’t a number. For example, you can multiply or divide it by 10 without it getting any larger or smaller. As far as numbers go, only zero functions that way.
What it boils down to is, in order for 1.0 and .999 not to be equal, there has to be a difference between them. There is no way in common math to express that difference, and if you try, it becomes pretty obvious the difference is zero.
This breaks down in so many ways. If 1 =/ .999… then either 1/3 =/ .333…, or 3(.333…) isn’t .999…
I can understand how a person who doesn’t think about it for long or doesn’t udnerstand math would think 1 and .999… aren’t equal, I didn’t think so when I first thought about it. But for Pete’s sake guys, that was high school.
Only at the limit of an infinite progression is there a limit. But since infinity is defined as limitless, there will always be a difference, even if there are no real numbers to express such a difference.
