The section “Filters on a Set” of the preprint of my math book is rewritten noting the fact that $\mathrm{Cor}\, \mathcal{A} = \uparrow^{\mathrm{Base} ( \mathcal{A})} \bigcap \mathcal{A}$.

In this blog I write mainly about mathematics. But this time I will allow myself to write on some philosophy from biblical Christianity positions.

Christ is truth, as it is clear from His own words: (John 14:6) “Jesus said to him, “I am the way, the truth, and the life.”

In my opinion, this means that Christ is the set of all true formulas. In other words, He is a boolean function which takes any formula and returns whether it is true.

Thus Christ is a perfect computer.

On the other hand, He is a part of physical reality: (Jn. 8:12) “… Jesus spoke to them, saying, “I am the light of the world…””, that is He is the electromagnetism or electromagnetic radiation of the Universe. But “light” is a part of the Universe.

So there is a real physical computer capable checking any mathematical formula for trueness.

Any formula! This means that anything which can be expressed as a formula really exists (and anything can be expressed as a set of math formulas some of which may be true and some are not).

Stressing this again: Any mathematical formula exists in reality.

This means, that laws of physics are not related to any particular subset of formulas, but anything expressed by any formula exists physically.

The conclusion of this is that the physical world and mathematical (platonic) world are the same!

The physics known to scientists is so just a part of mathematics. Our observable world is this way a part of mathematics (the platonic universe).

Another conclusion of this is, that the entire Universe is infinite and thus our observable meta-galaxy (and even the conjectured by physicists multiverse) is just a part of an infinite Universe.

I have proved that there is a bijection from the set $\mathsf{FCD}(A;B)$ to a certain subset of $\mathsf{RLD}(A;B)$ (which I call funcoidal reloids).

See section (currently numbered 8.4) “Funcoidal reloids” in the preprint of my book.

Just today I’ve got the idea of the below conjecture:

Definition I call funcoidal such reloid $\nu$ that

$\mathcal{X} \times^{\mathsf{RLD}} \mathcal{Y} \not\asymp \nu \Rightarrow \\ \exists \mathcal{X}' \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{X})} \setminus \{ 0 \}, \mathcal{Y}' \in \mathfrak{F}^{\mathrm{Base} ( \mathcal{Y})} \setminus \{ 0 \} : ( \mathcal{X}' \sqsubseteq \mathcal{X} \wedge \mathcal{Y}' \sqsubseteq \mathcal{Y} \wedge \mathcal{X}' \times^{\mathsf{RLD}} \mathcal{Y}' \sqsubseteq \nu)$
for every $\mathcal{X} \in \mathfrak{F}^{\mathrm{Src}\, \nu}$, $\mathcal{Y} \in \mathfrak{F}^{\mathrm{Dst}\, \nu}$.

Easy to prove proposition:

Proposition A reloid $\nu$ is funcoidal iff $x \times^{\mathsf{RLD}} y \not\asymp \nu \Rightarrow x \times^{\mathsf{RLD}} y \sqsubseteq \nu$ for every ultrafilters $x$ and $y$ on respective sets.

Conjecture $( \mathsf{RLD})_{\mathrm{in}}$ is a bijection from $\mathsf{FCD}( A ; B)$ to the set of funcoidal reloids from $A$ to $B$.

In the march of development of my theory, I have expressed many kinds of spaces (topological, proximity, uniform spaces, etc.) through funcoids and reloids subsuming their properties (such as continuity) for my algebraic operations.

Now I have expressed Cauchy spaces (and some more general kinds of spaces) through reloids. And yes, Cauchy continuity appears as a special case of continuous reloids (as expressed by my algebraic formulas).

My research related with Cauchy spaces is presented in this draft article.

When I first saw topogenous relations at first I thought that my definition of funcoids was plagiarized (for some special case).

But then I looked the year of publication. It was 1963, long before discovery of funcoids.

Topogenous relations are a trivial generalization of funcoids. However, I doubt whether anyone (except of myself) has defined $\langle f\rangle$ and $\langle f^{-1}\rangle$ for a funcoid, what allows to construct all beautiful theory of funcoids.

A famous mathematician Timoty Gowers asked this question: What is the difference between direct proofs and proofs by contradiction.

We, people, are capable of doing irrational things and to be discouraged.

I wrote an unthoughtful comment at Timoty Gowers’s blog. And this has caused Gowers to be discouraged and don’t continue to search an answer for his question. Yes, all of us, even Fields medalists such as Gowers are capable for doing irrational things.

On the other hand the irrational thing which I have done, was that I simply posted that somehow stupid comment despite of knowing that there are more about this (as described below). Note however that at the time of posting that comment I have not yet formulated the below exactly.

The trouble with my comment is that it applies only to the standard axiomatic theory of first order predicate logic. My comment is wrong for the way mathematics is usually written and is communicated. And this way differs of the customary formal predicate logic.

In customary informal mathematics the words “for all” and “exists” are not exact equivalent of ${\forall}$ and ${\exists}$ from first order predicate logic. The main difference is that we can have more than one proposition under a quantifier.

Examples:

For all ${x \in \mathbb{R}}$ we have ${e^x > 0}$ and therefore ${e^x > - 1.}$

In the standard first order predicate logic this would be written as two separate formulas:

${\forall x \in \mathbb{R} : e^x > 0}$; ${\forall x \in \mathbb{R} : e^x > - 1}$.

Have you noticed that in the informal mathematics we usually require ${x \in \mathbb{R}}$ only once and don’t write it twice as in the above formal formula?

Certainly we can invent a formalistic for this, like:

${\forall x \in \mathbb{R} : ( e^x > 0 ; e^x > - 1)}$.

Here we have multiple propositions under an universal quantifier.

It can be considered formally:

${\forall x \in U : ( P_0 ( x) ; \ldots ; P_n ( x))}$ is a shorthand for ${\forall x \in U : P_0 ( x) ; \ldots ; \forall x \in U : P_n ( x)}$.

Now an example about existence quantifiers:

There exists ${n \in \mathbb{N}}$ such that ${n > 2}$ and therefore ${n^2 > 4}$.

To formalize this, consider ${\exists n \in \mathbb{N} : n > 2}$ as a definition. After this formula we have ${n}$ defined and it must confirm to the proposition ${n > 2}$.

This could be written formally as:

${( \exists n \in \mathbb{N} : n > 2) ; n^2 > 4}$.

In customary logic this could be expanded to

${( \exists n \in \mathbb{N} : n > 2) ; ( \exists n \in \mathbb{N} : n^2 > 4)}$.

To formalize this after the ${\exists}$ definition we could introduce a new constant symbol (${n}$ in our example) and a new theorem (${n > 2}$ in our example). I leave as an exercise to describe the mapping from my formalistic (where ${\exists}$ denotes not just existence but also a definition) to the reader.

Now to the claimed topic of this post, the difference of proof by contradiction and a direct proof:

Suppose now we want to prove a proposition ${\forall x \in U : R ( x)}$.

Suppose ${\neg \forall x \in U : R ( x)}$. Then ${\exists x \in U : \neg R ( x)}$.

After this we can have a sequence ${( \exists x \in U : \neg R ( x)) ; P_0 ; \ldots ; P_n ; 0}$ of propositions using this ${x}$.

Or we can prove it directly: Then we would have a sequence

${P_0 ; \ldots ; P_n ; \forall x \in U: ( Q_0 ( x) ; \ldots ; Q_n ( x) ; R ( x))}$.

The choice of direct proof or proof by contradiction should depend on which of these two sequences of formulas is simpler.

Now suppose we want to prove a proposition ${\exists x \in U : R ( x)}$.

Suppose ${\neg \exists x \in U : R ( x)}$. Then ${\forall x \in U : \neg R ( x)}$.

Then we would have a sequence of propositions

${\forall x \in U : ( \neg R ( x) ; P_0 ; \ldots ; P_n ; 0)}$.

Or we can prove it directly: Then we would have a sequence

${(\exists x \in U: P_0 ( x)) ; P_1(x) \ldots ; P_n ( x) ; R ( x)}$.

The choice of direct proof or proof by contradiction should depend on which of these two sequences of formulas is simpler.

I leave open the following topic of discussion:

Show examples that sometimes a direct proof and sometimes a proof by contradiction is shorter.

Remark: All I have written here is about classic logic. I don’t deal with intuitionistic logic.