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I put this in the philosophy section but I guess it could equally go in a maths section being as I suppose it is the philosophy of maths.

Infinities are well defined in maths, I doubt anyone could disprove given the set of all natural numbers then the set of all fractions is 1 to 1 and countable and the set of decimals respectively 1 to many are uncountable.

Where this becomes more than maths though is in asking if the term infinity is really apt?

Now before you jump on my head and say of course it is! Cantor is a fricking genius you gnu!

Let me explain what I mean in philosophical terms:

An infinity cannot be mentally comprehended, the set [itex]\aleph{0}[/itex] for example is a symbolic representation that means the set of all natural numbers. It does not actually show all the natural numbers that would be impossible of course, it is merely an analogy or allusion.

Bearing this in mind then is there any reason to use the term infinity in this case, being as most people understand that infinity means an undefined quantity that is inumerable.

Can you diagonalise a concept that has no numerical representation? Are you allowed to define things as infinities in this manner? If you see what I mean by allowed. Is it philosophically "ok".

Is there a fundamental epistemological problem with calling [itex]\aleph{0}[/itex] an infinity when really its a sort of symbol of a value like [itex]\pi = 3.141...[/itex] is not actually pi but a hint at pi: at the limit of infinity the

[itex]\int^{\infty}_{0} \pi\; dx= \pi {x}+C[/itex]

That is it is exactly pi multiplied by a variable of x between 0 and infinity. of course we cannot draw a perfect circle because we cannot observe pi to the precision of infinity. But none the less we accept it has an exact value even if it doesn't physically exist per se.

If there is something greater or smaller than the limit of infinity though what does that say about limits and does it make any sense to use this outside of set theory, say in algebra, topology or calculus?

In short what use is Cantor's musings? And does it even make sense?

Infinities are well defined in maths, I doubt anyone could disprove given the set of all natural numbers then the set of all fractions is 1 to 1 and countable and the set of decimals respectively 1 to many are uncountable.

Where this becomes more than maths though is in asking if the term infinity is really apt?

Now before you jump on my head and say of course it is! Cantor is a fricking genius you gnu!

Let me explain what I mean in philosophical terms:

An infinity cannot be mentally comprehended, the set [itex]\aleph{0}[/itex] for example is a symbolic representation that means the set of all natural numbers. It does not actually show all the natural numbers that would be impossible of course, it is merely an analogy or allusion.

Bearing this in mind then is there any reason to use the term infinity in this case, being as most people understand that infinity means an undefined quantity that is inumerable.

Can you diagonalise a concept that has no numerical representation? Are you allowed to define things as infinities in this manner? If you see what I mean by allowed. Is it philosophically "ok".

Is there a fundamental epistemological problem with calling [itex]\aleph{0}[/itex] an infinity when really its a sort of symbol of a value like [itex]\pi = 3.141...[/itex] is not actually pi but a hint at pi: at the limit of infinity the

[itex]\int^{\infty}_{0} \pi\; dx= \pi {x}+C[/itex]

That is it is exactly pi multiplied by a variable of x between 0 and infinity. of course we cannot draw a perfect circle because we cannot observe pi to the precision of infinity. But none the less we accept it has an exact value even if it doesn't physically exist per se.

If there is something greater or smaller than the limit of infinity though what does that say about limits and does it make any sense to use this outside of set theory, say in algebra, topology or calculus?

In short what use is Cantor's musings? And does it even make sense?

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