Draw a circle on the X, Y axis with radius pi. All points on the circumference except 4 of them are irrational numbers. No others are rational, — EnPassant
There are "countably many" integers. That doesn't imply they can all be counted, but one can map a counting process to the set of integers. In the real world, that process would never end.How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19
There are "countably many" integers. That doesn't imply they can all be counted, but one can map a counting process to the set of integers. — Relativist
There are infinitely many numbers between 1 and 2. In fact, there are infinitely many real numbers between any 2 real numbers. This is the rationale for stipulating that there are "more" real numbers. It's not "more" in the real-world sense of your intuitions; it's "more" in a mapping sense. — Relativist
Infinity is not a thing that exists. It is a concept, and when it is applied to sets - it can lead to inconsistencies. There are infinitely many integers and infinitely many real numbers, but infinity is not a member of either set. Rather, "infinity" is a property of each of these sets. But is it the same property in both sets?Agree with all of the above. But you can't map one infinity to another with one being bigger than another because there isn't more than one. — Philosopher19
This is where I disagree. I don't believe Cantor's diagonal argument shows anything. Infinity is one cardinality/size, it makes no sense for one infinity to be bigger than another in terms of size.However, there is no 1:1 mapping between the reals and the integers. Reals map into integers, covering all the integers, but you can't cover all the reals with integers. This is the basis for saying the "size" of the set of reals is greater than the "size" of the set of integers. — Relativist
This is where I disagree. I don't believe Cantor's diagonal argument shows anything. Infinity is one cardinality/size, it makes no sense for one infinity to be bigger than another in terms of size. — Philosopher19
What's the basis for your claim that it makes no sense? — Relativist
In the everyday use of the term, a "quantity" is always a fixed, real number (e.g. a number of liters, a number of tomatoes, a number of molecules in a mole...). Infinity is not a real number. Your mistake seems to be that you're treating it as one.It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity. — Philosopher19
It makes no sense for one quantity of 10 to be bigger than another quantity of 10. 10 is one quantity. Similarly, it makes no sense for one quantity of infinity to be bigger than another quantity of infinity. Infinity is one quantity. — Philosopher19
In the everyday use of the term, a "quantity" is always a fixed, real number (e.g. a number of liters, a number of tomatoes, a number of molecules in a mole...). Infinity is not a real number. Your mistake seems to be that you're treating it as one. — Relativist
The definition of size here is the number of members. It is true that the number of members of an infinite set can never be specified, and the set is uncountable in that sense. (But we can confidently assert that the number of members - and hence the size - of an infinite set is larger than any finite set.)How would a difference in size be established between them when there is no counting involved? And if there is counting involved, how would infinity be reached? — Philosopher19
See Wikipedia - Transfinite NumbersIn mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers.
I'm suggesting that, in the face of the concept of infinity, there is more than one way to apply the relevant concepts. If we can choose one way rather than another, we cannot apply correct and incorrect. We need a different kind of argument.The mistake is to treat all of mathematics as a single system, with a single set of axioms and definitions. — Relativist
But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion?What matters is that there is a universe (the transfinite numbers) and that there are operations that can be performed with them - including a successor function for the transfinite ordinals - which allows treating them as greater than or less than.
It's still true that there is a conceptual relation between the transfinites and the reals and integers, and that was the basis for Cantor defining them. But it needs to be remembered that definitions (like "greater than" "less than" etc) are intra-system. — Relativist
Actually, because the reals and integer systems are applicable to the real world (they were developed by analyzing aspects of the real world), the terms "greater than" and "less than" do apply meaningfully.But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion? — Ludwig V
Agreed- it results in people treating infinity like a natural, or real, number. Then when non-mathematicians hear of transfinite numbers, it reinforces that false view - because it turns infinities into "numbers" but only in a very specialized sense.There is a constant tension here around the fact that counting cannot be completed and the temptation or desire to think of the infinite as some sort of destination or limit. — Ludwig V
To me, Infinity and Existence denote the same. — Philosopher19
But if definitions like "greateer than" and "less than" are only defined within a system, it follows that they cannot be applied outside it. Isn't that at least close to the OP's conclusion? — Ludwig V
Yes, I put that very badly. What I was getting at was that "largest number" or "smallest number" is not defined, or rather, the possibility is excluded by the definitions of "greater than" or "less than", or, more accurately, by the absence of any definition of "largest" or "smallest".Actually, because the reals and integer systems are applicable to the real world (they were developed by analyzing aspects of the real world), the terms "greater than" and "less than" do apply meaningfully. — Relativist
I'm afraid that, although I can see the sense of your conclusion, I do think your argument is mistaken at this point. My reason for accepting your conclusion is that infinity is not a number, so comparisons of size are meaningless.Infinity is one number just as 10 is one number. — Philosopher19
Well, umm.... in Zermelo-Fraenkel set theory infinity is taken as an axiom. Hence there's no proof for infinity.I’m a bit late to the thread. Just to put you at ease, mathematical infinities are not the same thing as philosophical infinities. They are precisely defined and used as means to perform mathematical equations. Which is pretty much what sime has said. — Punshhh
. . . so one could describe the situation like mathematicians have outsourced the philosophical problem to set theorists — ssu
Category theory would be the philosophers companion here, but uh... we haven't been trained in category theory in school or in the university. That is really something lacking!I like this. However, category theory - which includes categories of sets - an outgrowth of algebraic topology and what ever else of similar abstraction seem to have gotten into the game. — jgill
Category theory would be the philosophers companion here, but uh... we haven't been trained in category theory in school or in the university. That is really something lacking! — ssu
I remember someone saying that basically set theory was first seen as a way to finally solve the problematic nature of analysis.This has come up before. There are categories in my own subject of complex analysis, but in order to work with them you need a solid background of complex analysis at the beginning grad level. — jgill
I was a casualty of this "New Math" myself: on first grade they really started with set theory and believe me, for a first grader, it was indeed confusing. The old style with relating numbers to pieces of apples and toy cars with addition and substraction was far more understandable. I only remember how confusing "union of sets", "set substraction" and "intersection" was back then, because the teacher didn't give us any hint that somehow this was related to the old school addition and substraction. I also remember my grandfather and grandmother, both math teachers from my mother's side, having this negative attitude towards the new thing and talking with my parents and my other grandmother, that this is too difficult for a first grader.Category theory seems to be more a graduate school offering, whereas set theory can be presented at a much lower level. However, "New Math" of the 1960s and 1970s flopped when this was tried. Feynman was very critical of the effort. — jgill
if you mean by fractions rational numbers then there is but one transfinite cardinal (number) that counts them all, called aleph-naught. To understand this, do some research and learn. (That's the easiest way.) And along the way you will learn that aleph-naught is just the first and smallest of the transfinite cardinals, and that there are a lot of them. And not much of it intuitive. Cantor himself was apparently astonished to learn that the number of points on a line segment was the same as on a plane, or within any multi-dimensional space.an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions,.... — an-salad
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