Certainly there was no point in my scientific development where I started “communing with gods” or whatever. 🙂 Rather, first (as a teenager) I rediscovered stuff that was well-known since the 1700s or earlier to whatever extent it was correct, then I rediscovered stuff that was known since the 60s, then I rediscovered stuff that was known since a year or two previously, and finally I discovered stuff that turned out *not* to be known and to be publishable. Pretty minor stuff at first.

Was it that you knew more facts than other people? Or that you were somehow able to squeeze more juice out of the same stone? Did you saw things differently than other people, or were you faster, for whatever reason, at deriving an inevitable conclusion?

]]>Really all that it is reasonable to expect in life for the general case is plausible truth without counterexample. Truths that are mathematically certain are pearls to be treasured.

]]>Thank you for your observations. Thirty-seven years ago I began studying set theory using the expression,

$$\forall x \forall y (( x \subset y \leftrightarrow ( \forall z ( y \subset z \rightarrow x \subset z ) \wedge \exists z ( x \subset z \wedge \neg y \subset z ) ) )$$

for “subset” with membership given with

$$\forall x \forall y (( x \in y \leftrightarrow ( \forall z ( y \subset z \rightarrow x \in z ) \wedge \exists z ( x \in z \wedge \neg y \subset z ) ) )$$

I use the square subset now because this is better understood as an intensional part relation (tex not accepting sqsubset, not working for me). There is a lot of work involved with developing an extensional subset from this.

In these investigations, I realized that the second sentence can be changed slightly to accommodate “essential divisors” relative to “proper divisors.”

$$\forall x \forall y (( x \in y \leftrightarrow ( \forall z ( y \subset z \rightarrow x \in z ) \wedge \forall z ( z \subset x \rightarrow \forall w ( z \in w ) ) ) )$$

An essential divisor is either the unit or a prime. The membership relation is re-interpreted as essential division.

Number theorists study divisibility. And, divisibility relates to foundational debates in various ways. Around 2012 Leinster wrote blogposts on set theory and order. In so far as these sentences relate to ideas from both numbers and sets, they are candidates for investigation.

As you observed, math is hard. These sentences have made me question every claim coming from foundational mathematics. It is nice to see that others recognize the subtle problems within the hyperbole.

]]>The Shtetl-Optimized Committee of Guardians (SOCG) has been doing a wonderful job at talking me down from responding to things that I really shouldn’t! 🙂

]]>Are we going to discuss incel/nerds-and-sex issues on this blog ever again, or did that asshole kid ruin it for the rest of us?

Also, what’s the latest on that—did you get the incel, incels, wokes, or whatever the hell they were to stop trolling the Shtetl?

Scott B.

]]>Joshua’s #71 comment was interesting to ponder. I don’t think a diagonalization argument will work to prove quintic unsolvability, because we are comparing two countable infinities – 1) the set of polynomials that are solvable by radicals and 2) the set of algebraic numbers. If I am not mistaken, diagonalization arguments only work if we are comparing two different types of infinities.

Along the same lines, here’s a simpler problem to chew on: is there a diagonalization argument to prove that sqrt 2 cannot be expressed as a rational number?

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