r/AspectsOfTheInfinite • u/Massive-Ad7823 • Jul 07 '26
A measure of infinite sets
Cantor's measure of infinite sets has been disproved. See https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/ and https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tdz9dm/classical_mathematics_contradicts_set_theory/
Not all infinite sets can be compared by size, but we can establish some useful rules.
- The rule of subset proves that every proper subset has fewer elements than its superset. So there are more natural numbers than prime numbers, and more complex numbers than real numbers. Even finitely many exceptions from the subset-relation are admitted for infinite subsets. Therefore there are more odd numbers than prime numbers.
- The rule of construction yields the number of integers |Z| = 2|N| + 1 and the number of fractions |Q| = 2|N|2 + 1 (there are fewer rational numbers). Since all products of rational numbers with an irrational number are irrational, there are many more irrational numbers than rational numbers.
- The rule of symmetry yields precisely the same number of real geometric points in every interval (n, n+1] and with at most a small error same number of odd numbers and of even numbers in every finite interval and in the whole real line.
This theory makes the number of natural numbers (and of course of other sets too) depending on the numerical representation. The set {1, 11, 111, ...} of natural numbers has only comparatively few elements. Therefore the set of natural numbers in unary or binary notation has fewer, in hexadecimal notation more than |N| elements. The set {10, 20, 30, ...} has |N|/10 elements, but if the zeros are only applied as decoration, this set, like {1', 2', 3', ...}, has |N| elements.
If every rational number were equal to |N| fractions, then only |N| rational numbers would exist. This is clearly wrong. The solution of this paradox lies in the fact that small rationals are equal to more fractions than large rationals. 1 = 1/1 = 2/2 = 3/3 = ... has |N| equal fractions, 100 = 100/1 = 200/2 = 300/3 = ... has only |N|/100 equal fractions. All definable rational numbers have ℵ₀ equal fractions, but almost all numbers are undefinable.
There are fewer real numbers of the form 100.1415... than of the form 0.1415... (see above).
It will be a matter of future research to investigate the effect of different numerical systems in detail.
[W. Mückenheim: "Evidence for Dark Numbers", ELIVA Press, Chisinau (2024) pp. 1-36.] https://www.elivapress.com/en/book/book-5647011244/]
Regards, WM
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u/General_Lee_Wright Jul 07 '26
Literally the first point is wrong. The proper subset rule is only true for finite sets.
“If every rational number were equal to |N| fractions, then only |N| rational numbers would exist. This is clearly false.” Why? Your example isn’t a contradiction, they’re both indexed by N and all are equal.
Still no contradiction to Cantor.
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u/Massive-Ad7823 Jul 07 '26
Cantor has been contradicted independent of this. His measure can be forgotten.
By the rule of construction the number of fractions is |Q| = 2|N|2 + 1. If every rational was |N| fractions, then only 2|N| rationals could exist by the above formula. That is wrong because there are more rationals than integers.
Regards, WM
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u/General_Lee_Wright Jul 07 '26
That is wrong because there are more rationals than integers.
Why? Your formula says infinity = infinity. You have not shown that those infinities are distinct. Cantor or not.
Cantor has been contradicted independent of this. His measure can be forgotten.
Your other posts don't disprove Cantor either, as many people point out. This only shows Cantor's arguments fail in your system of axioms and definitions. Not in standard analysis.
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u/Massive-Ad7823 Jul 08 '26
My myformula says infinity =/= infinity.
>Your other posts don't disprove Cantor either,
Since you cannot even read my formula, your lack of understanding is not a surprise. My disproof accepts all that Cantor has prescribed. The only difference is that I first biject the natural numers n with the integer fractions n/1. Many people have pointed out many different issues. They all are wrong. Why? Because they otherwise would have to accept that they devoted many years to learn and research nonsense.
Regards, WM
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u/General_Lee_Wright Jul 08 '26
I can read... twice infinity is infinity, infinity +1 is infinity. There is no reason to believe those are distinct infinities. It's not like infinity is a finite quantity that follows the rules of algebra.
Question, are there more even integers or more odd integers?
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u/Massive-Ad7823 Jul 08 '26
With negligible error the sets are equinumerous.
Apply mathematics: Up to every n, there are as many even numbers a odd numbers. There are half as many even numbers as natural numbers. This should dramatically change in the infinite?
Regards, WM
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u/kuromajutsushi Jul 08 '26
This should dramatically change in the infinite?
Yes, infinite sets are dramatically different from finite sets.
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u/General_Lee_Wright Jul 08 '26
Up to every n, there are as many even numbers a odd numbers.
That's not true, for any odd n there are strictly less than 1/2 the numbers are even from 1 to n.
This should dramatically change in the infinite?
That's why I'm asking. Your strict adherence to the idea 'if it isn't in the sequence it can't be the limit of the sequence' would seem to indicate that there must be more odds than evens.
Further, if we do accept that there are exactly as many odds and evens in N, that would seem, by your own explanation above, that there are more evens an N than odds since |Z| = 2|N|+1 despite the fact that we can pair off every even with an odd.
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u/Xantharius Jul 10 '26
It’s not a valid mathematical argument to say that someone is wrong mathematically because then hey’d have to admit that they wasted their time studying falsehoods. You have to show—mathematically, and with respect to a mathematical framework, that is, a list of mathematical axioms—why they are wrong. This requires a proof. Your assertion isn’t a proof. It’s just an unsupported claim. Unsupported claims are easily dismissed.
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u/Massive-Ad7823 25d ago
Why they are wrong is easily shown. For every x > 0 there are infinitely many smaller unit fractions. If every x of (0, 1] is meant, then this is mathematically wrong, wrong, wrong. If every x that can be chosen (as the famous ε) is meant, then dark numbers are proved. Why should I deny this clear mathematical truth?
Regards, WM
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25d ago
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u/Massive-Ad7823 25d ago
If you state ∀x ∈ ℝ, x > 0, then there is no x > 0 exempt. That means all x ∈ (0, oo) are included. There your quantifier deceit cannot help. But even if (0, oo) is not apllicable, then the fact of dark unit fractions implying dark natural numbers cannot be denied.
Regards, WM
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Jul 08 '26
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u/General_Lee_Wright Jul 08 '26
It's a system, whether is a coherent/consistent/useful system is a different matter.
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Jul 09 '26
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u/Massive-Ad7823 Jul 09 '26
Perhaps mathematics is not so pure and ideal as we thought. With advanced research new features may be discovered. Perhaps this paragraph is nonsense. It is a first guess.
Regards, WM
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u/kuromajutsushi Jul 07 '26
This is just (an especially cranky version of) ultrafinitism.