Define the class of functions |P to be any function P(x) that has these two properties:

(1) If x is in the range of M(*), then so is P(x).

(2) M(P(x))=M(x)+1.

The Principle of Induction says that any such function, with at least one object in its range, defines an infinite set. The issue here is whether this principle should be accepted. The inconsistencies WM claims to find are the direct result of comparing a Mathematics that accepts it, and makes logical inferences from it, to one that denies it.

The “final result” of applying the P(x) that defines Zeno’s paradox, to all of the intervals in the set it defines, must exist. This can be proven in any mathematics that can define the intervals (notice I don’t claim a “universal” Mathematics, as WM does). That is, Achilles *CAN* catch Tortoise. But it can also be proven that this “final result” cannot be a member of the set itself. So a consistent Mathematics must be able to define a “final result” of P(x) for an infinite set, that is not in the set.

According to WM’s argument about rows, if there is a “final result” of applying the P(f) that defines the set |F of all FISONS, then it must be a FISON. But if it is a FISON, then inclusion monotony is violated.Therefore, he claims, any mathematics that produces such a “final result” is inconsistent.

WM’s conclusion “it must be a FISON” simply doesn’t follow, because he does not define what “final result” means for an infinite set. The inconsistency he finds is an inconsistency in any Mathematics that does not have such a definition. That is, WM’s “Universal Mathematrics” is inconsistent because it does not include Set Theory’s Axiom of Infinity. The contradiction he claims is actually proof that he needs to supply a definition, like the one Set Theory defines. The one where Aleph_0 is the “final result” of applying P(n)=n+1 to all of |N, but is not in |N.

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be “bigger” than any row? Or – as is the case – is there a problem defining “the maximum over all rows” and “the union of all rows” when the set of rows is infinite? And so does using the expression require you to step outside of your very narrow box, and accept that there is no finite definition for either?

But I see that simple mathematical arguments about infinite sets, because they are based on principles in mathematics that you do not want to accept, are useless in your case.

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cannot be longer than all rows. This is simply prohibited by definition and is independent of any axioms or other conditions. 99 % of all intelligent humans understand that by shifting all numbers of all infinitely many rows into the first row this first row cannot become longer than all contributing rows which are finite.

Knowing that this proves my case I will stop here. Remain happy with your delusions. They will not cause damage because set theory is total nonsense and therefore without any effect to real life.

Regards, WM

]]>No, it is you who doesn’t understand.

|N is an inductive set, defined by two rules. (1) It includes the number 1, and (2) if it includes the number n, it also includes the number n+1. The second part defines every member of IN after 1, even though you can’t apply it, pairwise, to define every member. Any set you have tried it on implies one more element you haven’t.

Another way to say that is that the definition process continues toward *POTENTIAL* infinity, but can’t reach an end (which is what “potential” means here.) But the Axiom of Infinity allows us to treat the set as *COMPLETED* by this inductive step. I call the the principle of induction: even though you can’t reach the end of what it defines, it still defines all possible elements.

This principle applies to any pairwise operation. We can’t pairwise-add every term in a converging sequence like 1/2+1/4+1/8+ … .But the sum *OVER* the entire set can be determined. AND IT IS A VALUE THAT CAN’T BE FOUND BY PAIRWISE SUMMATION.

The maximum *OVER* a converging sequence like that in Zeno’s Paradox, my {X1, X2, X3, …}, also exists. You claim, *CORRECTLY*, that this maximum isn’t a member of the set itself. But the Axiom of Infinity still says that the maximum *OVER* the set exists. Any mathematics that denies this existence is self-contradictory, since it also says that Achilles catches Tortoise at that maximum, and that it happens at X=X1*VA/(VA-VT).

Your |F is also an inductive set, defined by two very similar rules. (1) It includes the FISON {1}, and (2) if it includes the FISON {1,2,…,n}, it also includes the FISON {1,2,…,n,n+1}. The pairwise union of elements always produces a FISON that is equivalent to the largest FISON used so far. But the union *OVER* the *ENTIRE* set, which can only defined by accepting the Axiom of Infinity and the principle of induction, is defined to be the set |N, which is not a FISON.

THIS IS NOT A CONTRADICTION. It is a logical consequence of accepting the Axiom of Infinity, and the principle of induction. BUT, any mathematics that does not accept them does have a contradiction, as described by Zeno of Ilea.

]]>No, I don’t. If the union of all FISONs has more naturals than any FISON then these naturals must be in the FISONs to be unioned already. Therefore your distinction is nonsense.

JJ> This is no more a contradiction than “The set of all natural numbers has cardinality greater than any natural number.”

No, you have not yet understood. Try again: If all FISONs (or the unions of all FISONs) contain more naturals than any FISON, then inclusion monotony is violated. Hence mathematics is violated.

Regards, WM

]]>A FISON is not a natural number, it is a set of natural numbers. Set Theory does not “require that the set of all FISONs” contain any natural numbers, only that it contain sets.

I think you mean “the *union* of all FISONs, has more natural numbers than any FISON.” This is no more a contradiction than “The set of all natural numbers has cardinality greater than any natural number.” In fact, the two statements are equivalent, and both are accepted by the Axiom of Infinity. Withit, there is no contradiction here.

What you are ignoring, it that it is your unspecified, and non-existent, conglomeration of different fields of mathematics that has the contradiction you keep finding. Specifically, there is one derivation that shows that Achilles catches Tortoise at X=X1*VA/(VA-VT). There is another that says he catches up only at the “greatest” member of the set {X1,X2,X3,…}. This is a monotonically increasing sequence that, according to you, can have no greatest member.

This is your contradiction. All forme you have found can be shown to be equivalent to it. It is in your ill-described mathematics, no matter how you obfuscate it. The mathematics that excludes the Axiom of Infinity. The Axiom that says what you call a contradictory statement is, in fact, an accepted truth.

]]>The conclusion follows from mathematics. Set theory requires that the set of all FISONs has more natural numbers than any FISON. That is contradicted by mathematics, in particular by the well-ordering of FISONs.

Definition: A set of small FISONs is a set of FISON the union of which is one of the FISONs. Example {{1}, {1, 2, 3}, {1, 2, 3, 4, 5, 6, 7}} is a set of three small FISONs.

If there are sets of FISONs that do not obey this rule, then there must be a first FISON violating it. Find it! Or stop claiming that the set of all FISONs can have a union greater than every FISON.

JJ> I ask you: What set of axioms are you using that allow you to express the concept of actual infinities?

Actual infinity is tantamount to the claim quoted above: The union of all FISONs is |N which is a larger set than every FISON.

JJ> Wm also said: “Using analysis … the limit …” What axioms support saying the limit is part of the interval?

In analysis the limit of a strictly monotonically increasing sequence is never a term of the sequence. The union of an inclusion monotonic sequence is always a term of the sequence.

Therefore:

The set theoretical limit of the sequence of intervals is its union: lim[0, 1 – 1/n] = [0, 1).

The analytical limit of the sequence of intervals: lim[0, 1 – 1/n] = [0, 1].

Both differ like:

The set theoretical limit of the sequence of FISONs lim(F_n) = U(F_n) = |N.

The analytical limit of the sequence of FISONs lim(F_n) = oo or omega or aleph_0.

In the latter case set theory claims |N = omega or Card|N = aleph_0.

This is obviously wrong because the union of FISONs cannot be an actual infinity like the union of intervals [0, 1 – 1/n] cannot be [0, 1].

Regards, WM

]]>But, to address the point you think you were making: There is no infinite natural number. But the set of all natural numbers is infinite; i.e., it has an infinite cardinality. THIS IS NOT A CONTRADICTION IN ANY MATHEMATICS. Any mathematics that includes the Axiom of Infinity takes it as a given. Any mathematics that doesn’t cannot express the concept of something being infinite.

There is a bijection between natural numbers and FISONs, where each natural number n is uniquely paired with the FISON F(n) whose cardinality is n. By definition, every natural number is included in this bijection. By definition, no FISON is missing from this bijection. So, just like every natural number is finite, every FISON has a finite cardinality.

Any two sets in a bijection with each other have the same cardinality, The set of natural numbers |N has infinite cardinality, but contains no member that is infinite. The set of FISONs |F has infinite cardinality, but contains no member with infinite cardinality.

If you want to try to rebut this, please say which of the preceding statements you think is wrong.

WM continued: “… then the union of all FISONs (or all natural numbers) has not an actual infinity of elements.” There is no infinite number n, but the Axiom of Infinity says the union of all natural numbers has an “an actual infinity of elements.” So your claim that the premise “Set X has no infinite elements” leads to the conclusion “set X has not an actual infinity of elements” is an example of the logical fallacy known as “affirming the consequent.” The conclusion does not follow from the premise.

Again, any set of Axioms that accepts an Actual Infinity says the opposite of your conclusion is what is correct; and any set that does not accept an Actual Infinity cannot express the premise or the conclusion. So again, and more specifically, I ask you: What set of axioms are you using that allow you to express the concept of actual infinities?

Wm also said: “Using analysis … the limit …” What axioms support saying the limit is part of the interval?

]]>You can understand this better in a comparable finite case: In set theory the sequence of closed intervals [0,(n−1)/n] has as its limit the half open interval [0,1). Using analysis we obtain 1 as the limit of the right-hand side borders (n−1)/n. Connecting every point (n−1)/n to the origin 0 cannot change the limit. Therefore the analytical limit is the closed interval [0,1].

Apply this to the limit omega of the sequence and to the union |N of all FISONs, then you might see the fundamental mistake which set theory is based upon.

Regards, WM

]]>“Cantor uses curly brackets for wsets and parentheses for ordered sets,” Not in the proof in question. And regardless, that is just a notational convention. One you do not support in your book, so this is irrelevant.

“In order to understand the argument about FISONs note that every set of FISONs, when unioned will yield one of the FISONs,” And any two natural numbers, when provided as arguments to a “maximum” function, yields a natural number. This is a perfect analogy, and holds for all natural numbers. Again, you argue with inconsistency, and irrelevancy.

Nobody claims the set of natural numbers should contain an infinite number. Similarly, there is no reason to think an infinite set of FISONs should contain an infinite FISON, or a “FISON violating inclusion monotony.”

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