My Math Blog

A Modification of Naive Set Theory

Posted on September 4, 2025

We frame this in first-order predicate logic with a binary relation "∈", as in Naive Set Theory (NST).

Let

φ

be a unary predicate and

p

the proposition:

\exists y \forall x (x \in y \Leftrightarrow φ (x))

Then:

If we know that $p$ is true, we can form the set $M = {x | φ (x)}$ .
If we know that $p$ is false, we cannot form the set $M$ .
Otherwise, we say that the existence of set $M$ is a hypothesis and we can investigate its consequences in theory.

Note:

p

and

M

depend on

φ

Approximation of total natural functions through primitive recursive functions on finite sets

Posted on August 28, 2025

In this post we state that if $f : ℕ \to ℕ$ is a total function, and $A \subseteq ℕ$ is a finite set, then $\exists g : ℕ \to ℕ$ a primitive recursive function such that: ${f |}_{A} = {g |}_{A} . ∎$

On the non-existence of Russell's set

Posted on August 28, 2025

In this post we prove that $\neg \exists R \forall x x \in R \Leftrightarrow x \notin x .$

Indeed, $\forall R \exists x x \in R \Leftrightarrow x \in x,$ because $\forall R R \in R \Leftrightarrow R \in R . ∎$

A Divisibility Criterion with 41

Posted on August 27, 2025

In this post we write natural numbers in decimal.

Let $n$ be a 100-digit natural number.

Let us form 20 of 5-digit numbers with the digits of $n$ in the order in that they appear in the decimal representation of $n$ (left to right).

Then $n$ is a multiple of 41 if and only if the sum of the 20 numbers mentioned above is a multiple of 41. (This works because $10^{5} \equiv 1 (mod 41)$ .)