mmmhhh. It’s clear: My paper holds a new approach and it is more a kind of experimental mathematics. But the aim is to start a discussion about the basics of set theory. The existence or not existence of isolated prime numbers – as I defined that prime subset – is not provable with a finite set of terms/ forms because we need infinite many steps to calculate the properties (having not a prime cousin partner).

Did it was unwise to read about this? It was your decision to think that. And it was my decision to wrote it and a decision of a spacialist about Gödel theorem endorsing my first paper…

Hoping reading my second paper about a substitution of Brouwers choice sequence makes clear whats going on.

Thank you very much and best regards,

Klaus Lange

NS

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“When in doubt, interchange the order of integration and summation. Miracles can occur.”

I wonder if it’s already on the wiki?

In response to the above remarks of Daryl McCullough, let me say what my instincts are. In general they’re pretty similar to Matt’s: I am uneasy about any notion of mathematical truth that smacks of metaphysics. However, I have degrees of unease, and I’m not sure I can justify them. I can see that I would have a hard time defending the view that there is no fact of the matter as to whether every even natural number is the sum of two primes (jokes about 2 being a counterexample apart), but if I go one level deeper in the quantification then my attitude changes.

For example, is there a fact of the matter about whether the decimal expansion of contains infinitely many 0s? The difference here is that an imaginary thought experiment (in which, say, you are punished for your earthly sins by being put in a cell and told to watch as the digits of run past you for ever) never comes to an end, regardless of what you observe at any finite stage. The analogous Goldbach verification probably wouldn’t come to an end either, but one can at least entertain the possibility that an even integer might come along and spring a huge surprise.

One might argue as follows. If I concede that there is a fact of the matter as to whether contains at least one 0 after the nth digit, then surely I am now reduced to asking whether some factual statement holds (namely not having any more 0s after the nth digit) for at least one And now I am back with a statement of the Goldbach type. But I don’t buy that, because in the Goldbach situation one can check statement in finite time and here one cannot: it seems to me to be genuinely worse.

If we deepen the quantification further, by asking whether the proportion of 0s in the decimal expansion of tends to 1/10, then all these concerns only increase. The imaginary thought experiment becomes wilder and wilder.

In fact, I have some sympathy with ultrafinitism: I’m tempted to say that there may be no fact of the matter as to whether the proportion of 0s in the first Ackermann(10,10) digits of is under 20 percent. Of course, if someone came up with a proof, then I would change my tune completely. But what if it was a statement that had no proof short enough to write down?

I do have one argument on the other side, which is that although we may not have a proof, we do at least have evidence that we can look at, such as how the digits of behave. The fact that we take this evidence seriously could be taken as showing that we do have some concept of the truth or falsity of statements about the long-term behaviour: it’s just that to justify our beliefs we use inductive rather than deductive methods. But this is to use the word “truth” in a somewhat different way.