So many people say “infinite thing is not a number”, but we are able to take very freely all the infinite things (such as infinities and infinitesimals) as numbers in the field of analysis and set.
The question is not well posed. "Infinities" normally refer to cardinalities. It is known that if A is infinite set, then P(A) is a strictly bigger infinite set. This yields infinitely many infinite cardinalities.
Infinitesimals refer to elements of a non-archimedean ordered field, which are in absolute value smaller than every positive rational number. Their multiplicative inverses are "infinitely big elements" - but not infinities. For n in N, if e is infinitesimal then ne is infinitesimal; respectively if B is infinitely big, then nB is infinitely big. So there are infinitely many infinitesimals and infinite many infinitely big elements in the non-archimedean ordered field.
In both cases there is really nothing special. Just a little bit of mathematics.
The question is not well posed. "Infinities" normally refer to cardinalities. It is known that if A is infinite set, then P(A) is a strictly bigger infinite set. This yields infinitely many infinite cardinalities.
Infinitesimals refer to elements of a non-archimedean ordered field, which are in absolute value smaller than every positive rational number. Their multiplicative inverses are "infinitely big elements" - but not infinities. For n in N, if e is infinitesimal then ne is infinitesimal; respectively if B is infinitely big, then nB is infinitely big. So there are infinitely many infinitesimals and infinite many infinitely big elements in the non-archimedean ordered field.
In both cases there is really nothing special. Just a little bit of mathematics.
Thank you for the frank ideas.
Just see the following fact:
We say the infinity of real number set is bigger than the infinity of natural number set because the elements in real number set are more than those in natural number set------- the elements in natural number set are finished, limited, finite, no-endless while the elements in real number set are unfinished, unlimited, infinite, endless.
The operation of “one-to-one correspondence” of the elements between the two sets has clearly proved that infinite real number set is infinite but infinite natural number set is finite.
So, what is finite (finished, limited, no-endless) and what is infinite (unfinished, unlimited, endless)?
Infinite natural number set is in fact finite!
This is the very reason for the unavoidable Russell’s paradox family.
For Chris Cook:
There is however a mathematical construction which contains in the same structure, which is a real closed field, both the real numbers and all ordinals. This is the field NO of Conway. See his book "On numbers and games". There you have well defined objects like \omega x \pi, and like pi x 10^\omega. Also there \omega + 1 = 1 + \omega. The subject has been further developped by Gonshor and many other authors, also in journal articles. And of course in this field 1/ \omega is an infinitesimal!
If one constructs the Robinson infinitesimals, then one can see that as Mihal said, this is an non-Archimedean ordered field. Essentially it is a non-Archimedean non-complete linear continuum. In it there are smaller/bigger infinitesimals and unlimited corresponding numbers. In order to appreciate all these, one should construct or read such a construction. Unfortunately I have one, but it is in Greek. I post it since even the pictures may say something to you!
It is great to share your frank ideas. Just see the following fact:
Nothing can run away from the “potential infinite--actual infinite” theatrical frame within present classical infinite related science and mathematics, not matter standard or non-standard or some future new “former languages”, they bear all the defects and errors disclosed by the suspended thousands-year-old “potential infinite--actual infinite related paradox families”---------because non of such “potential infinite--actual infinite” related theories can answer this basic question: for any infinite related “numbers (numerical mathematical things)”, are they potential infinities (infinitesimals) or actual infinities (infinitesimals)?
In front of “potential infinite--actual infinite”, non-standard infinities (infinitesimals) are nothing new to standard infinities (infinitesimals).
It is very easy to talk about what infinities and infinitesimals are theoretically but it is impossible to treat them in some practical situations as what is talk about, for example:
1, Are those Un--->0 items in Harmonic Series infinitesimals, why?
2, If they are not, what are they and what is infinitesimal?
3, If they are, are they potential infinitesimals or actual infinitesimals, why?
4, Can we produce infinite items each bigger than 1/2, or 100, or 1000000, or 1000000000000000000,…by brackets-placing rule from Un--->0 items in Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 100000000000000000000000000000)?
Since antiquity, no one in the world with the standard or non-standard or some future new “former languages” within the theatrical frame of present classical “potential infinite--actual infinite” related science and mathematics are unable to answer such questions self-justificationaly!
Five of my published papers have been up loaded onto RG to answer such questions:
1，On the Quantitative Cognitions to “Infinite Things” (I)
https://www.researchgate.net/publication/295912318_On_the_Quantitative_Cognitions_to_Infinite_Things_I
2，On the Quantitative Cognitions to “Infinite Things” (II)
https://www.researchgate.net/publication/305537578_On_the_Quantitative_Cognitions_to_Infinite_Things_II
3，On the Quantitative Cognitions to “Infinite Things” (III)
https://www.researchgate.net/publication/313121403_On_the_Quantitative_Cognitions_to_Infinite_Things_III
4 On the Quantitative Cognitions to “Infinite Things” (IV)
https://www.researchgate.net/publication/319135528_On_the_Quantitative_Cognitions_to_Infinite_Things_IV
5 On the Quantitative Cognitions to “Infinite Things” (V)
https://www.researchgate.net/publication/323994921_On_the_Quantitative_Cognitions_to_Infinite_Things_V
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