The section “Filters on a Set” of the preprint of my math book is rewritten noting the fact that .

I have proved that there is a bijection from the set to a certain subset of (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 that

for every , .

Easy to prove proposition:

**Proposition** A reloid is funcoidal iff for every ultrafilters and on respective sets.

**Conjecture** is a bijection from to the set of funcoidal reloids from to .

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.

Read my book before.

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 and for a funcoid, what allows to construct all beautiful theory of funcoids.

This my post is about mathematical logic, but first I will explain the story about people who asked or answer this question.

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 and from first order predicate logic. The main difference is that we can have more than one proposition under a quantifier.

Examples:

For all we have and therefore

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

; .

Have you noticed that in the informal mathematics we usually require only once and don’t write it twice as in the above formal formula?

Certainly we can invent a formalistic for this, like:

.

Here we have multiple propositions under an universal quantifier.

It can be considered formally:

is a shorthand for .

Now an example about existence quantifiers:

There exists such that and therefore .

To formalize this, consider as a *definition*. After this formula we have defined and it must confirm to the proposition .

This could be written formally as:

.

In customary logic this could be expanded to

.

To formalize this after the definition we could introduce a new constant symbol ( in our example) and a new theorem ( in our example). I leave as an exercise to describe the mapping from my formalistic (where 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 .

Proof by contradiction:

Suppose . Then .

After this we can have a sequence of propositions using this .

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

.

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 .

Proof by contradiction:

Suppose . Then .

Then we would have a sequence of propositions

.

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

.

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.

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