Cantor avoids the paradox simply by having larger and larger infinities and not referring to the set of sets. — ssu
The paradox is rigorously avoided also by the axioms of ZF-logic. Some of the axioms are there basically only to deal with the paradox. — ssu
Well… what is your definition of a number? Numbers you see are used to measure something and when you have something that isn't measurable / countable, you have bit of a problem.A reasonable, working definition of infinity:
‘A number bigger than any other number’ — Devans99
Well, I'm a proponent of Absolute Infinity, but before going into that, a question:It is clear then that there can only be one such number - if there was a second infinity then both would have to be larger than the other. — Devans99
Well… what is your definition of a number? Numbers you see are used to measure something and when you have something that isn't measurable / countable, you have bit of a problem — ssu
So what do you then think about Cantor's finding that there are more real numbers than natural numbers? Or said another way, that you cannot put into 1-to-1 correspondence the real numbers with the natural numbers, as you can put the rational numbers with the natural numbers? — ssu
Sets are just ideas. They're something we made up. Can we make up ideas that run into consistency problems? Sure. And then we can make up modifications or restrictions to avoid the consistency problems. — Terrapin Station
But if you start with clean ideas (non contradictory axioms) you get clean theories. — Devans99
Wrong. They do. The cardinality of aleph-null and aleph-1 is not the same.Infinite sets have no cardinality. — Devans99
Really?I do not agree with the bijection procedure as a valid way to compare two things. — Devans99
Because the proof is a reductio ad absurdum proof.How can a procedure that is meant to demonstrate equality produce such obviously wrong results? It is because infinity has no size (it is unmeasurable) so it is impossible to compare the size of infinite sets. — Devans99
So now you are dismissing totally set theory. Good luck with that.It has no size. Infinite sets do not have a cardinality. — Devans99
So the bijection 1+2=3 you don't agree with? — ssu
So now you are dismissing totally set theory. Good luck with that. — ssu
Ok. Is infinity bigger than 54? Does 54 have size? No?
If 54 has size, then where does infinity loose it? — ssu
Well, you simply have to prove it in mathematics. If you show that either all or some the axioms of ZF are incorrect, then that is that's a positive breakthrough.What has been done in set theory is an abomination to the principles of sound design — Devans99
Does the number 54 exist in reality? Show me where the real 54 is.Infinity only exists in our minds, not in reality. — Devans99
REALLY? You think that defining something in math is something like 'writing it down'?It is never possible to fully define an infinite set - there is not enough paper in the world — Devans99
Well, you simply have to prove it[/i] in mathematics. If you show that either all or some the axioms of ZF are incorrect, then that is that's a positive breakthrough. — ssu
Does the number 54 exist in reality? Show me where the real 54 is. — ssu
Besides, I think infinity is used a lot in math and is a very useful, very logical mathematical object, which is inherent to mathematics in order for it to be logical. — ssu
REALLY? You think that defining something in math is something like 'writing it down'? — ssu
Stones can exist. Yet Again you have the same illogical idea here: two googolplex of stones cannot exist. And where in reality exists this '54'?54 stones can exist. An infinite number of stones cannot. — Devans99
Congratulations! You've made it to Aristotle with accepting potential infinity.Potential Infinity (limits in calculus) is useful. — Devans99
The whole error is then to deny the existence of the paradoxes and think that everything in mathematics is fine and dandy if we a) don't approach this question or b) ban it.Set theory is rife with paradoxes because of infinity. — Devans99
Stones can exist. Yet Again you have the same illogical idea here: two googolplex of stones cannot exist. And where in reality exists this '54'? — ssu
The existence of the paradoxes show simply that our understanding of infinity is still lacking. — ssu
At this level, it is estimated that the there are far less than a googol of atoms in the observable universe. As stones consist of more than one atom, obviously two googolplex of stones cannot exist.I'm not sure what you mean? Two googolplex of stones can exist IMO. — Devans99
I agree with this. We makes assumptions that are contradictory to each other. So what are we lacking? That's the interesting question.My view is that paradoxes indicates that there is an error in the explicit/implicit assumption underlying the problem. — Devans99
I have to disagree with you in this one. The set of natural numbers N does exist in the Mathematical realm. It is an infinite set as it surely isn't a finite set of numbers.A paradox is just a contradiction so it is a form of proof via contradiction that infinite sets do not have sizes / infinity does not exist. — Devans99
Beware the words 'clearly' and 'obviously'. When used, they are nearly always wrong. That is the case here. If you think otherwise, try to prove that a set of all sets exists! — andrewk
At this level, it is estimated that the there are far less than a googol of atoms in the observable universe. As stones consist of more than one atom, obviously two googolplex of stones cannot exist. — ssu
I have to disagree with you in this one. The set of natural numbers N does exist in the Mathematical realm. It is an infinite set as it surely isn't a finite set of numbers. — ssu
It's fine for you to do that. But realise that most people do not share your opinion, so their beliefs will differ from yours. From what I have seen of your posts on infinity, the paradoxes you think you see stem from that belief, so they are not paradoxes for other people.I still maintain that infinity is unmeasurable so has no size - that is the real cause of most of the paradoxes of infinity. — Devans99
I still maintain that infinity is unmeasurable so has no size — Devans99
So now it would be possible. But it isn't possible.OK fair point, but my meaning was if sufficient stones existed, the a googolplex of stones would be possible. — Devans99
Numbers exist in our heads only. And likely some animals use a mathematical system of "nothing, 1,2,3, many.) which is a totally functional system if you don't have an issue with or the need to count to something more than three. So likely this whole system of counting isn't only limited to humans. Yet in the physical realm there is no number 54. 54 doesn't exist physically. So it doesn't exist.The set of natural numbers exists in our heads only — Devans99
Infinity has the size of infinity. An infinity. One infinity.I still maintain that infinity is unmeasurable so has no size - that is the real cause of most of the paradoxes of infinity. — Devans99
Sorry to butt in like this but I think the correct term is interminable and not unmeasurable. The difference is that the former captures infinity as a quantity while the latter seems to treat infinity as a quality. — TheMadFool
Infinity is not a number/quantity: — Devans99
We don't have to worry about its beginning but it seems to me that it'll extend into the infinity of the future — TheMadFool
Do you think this self-reference is important? Does it result in the paradoxes we see in the math of infinities? — TheMadFool
You can resolve that by considering eternity isn't bigger than itself; so it's finite. — Shamshir
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