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Using theories and proofs. Why couldnt there be an end to numbers?

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Absolute Infinity was "conceived" or lets say discovered by Georg Cantor and was considered to transcend all the transfinite numbers and the the set of all sets (paradox). But there was never any explanation to why there couldnt be an end to numbers, to humans, to whoever. Now this might sound like a stupid thread, i might be given explanations "because it just goes on and on and on" well that is true but we couldnt reach Absolute Infinity just by going on and on. Why arent the line of numbers with a start and an endpoint but the "space" of numbers in between the start and end is continuously infinite, transfinite, or Absolutely Infinite? Why isnt there a "Totality" that goes beyond all conceivable, inconceivable numbers while everything below "Totality" "just goes and on and on and on" ?

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eg.

 

a=startpoint of numbers

b=endpoint of numbers

 

a <-------------------------------------------------------------------> b

 

^ this area in between a and b is continously infinite.

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I dont quite understand your question ?

But there is a simple set theoretic construction allowing you to add a maximal element to R, the set of all real numbers (keeping well ordering and all)

 

Mandrake

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Let R be the largest possible number.

 

Let S = R + 1.

 

S > R, hence R cannot be the largest possible number, hence there cannot BE a largest possible number.

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But lets say the largest possible number is absolutely infinite by properties.

 

R + 1 = S

 

but...

 

S = R

 

so...

 

R + 1 = R

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If you can add one to it in an operational sense, doesn't that make R "not infinite"?

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Exactly. Infinite is not a number, it's an adjective.

I.E: Infinite + 1 cannot exist, since infinite is not a number.

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Buy a book on analysis. The real numbers are closed under additon so for any x, x + 1 is also a number of the rela numbers, further x < x + 1, therefore there is no largest real number (of course it's not enough just to say this, it must be proven from the axioms of the real numbers, which is why I suggest you buy a book on mathematical analysis).

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Did I not just describe it? I just said that R is an infinite number.

In that case you are certianly not talking about the real numbers.

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Did I not just describe it? I just said that R is an infinite number.

Were you talking to me, or Refsmmat?

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