In details

The second truth

Let's look a little at the first truth of Zermelo-Fraenkel set theory.

ZF (1) Extension Axiom:

if The and B are sets and if for all x The if and only if x B, then The = B. Economically (in mathematical symbols):

It is very important for you to understand the "grammatical" structure of the statement above. In words, there it is written that for whatever set x of the universe of sets that we don't know yet, if the fact that it belongs to the set The is equivalent to the fact that he belongs to the Bso it is worth saying that a = b. This means that from now on, whenever someone claims that a = b, the meaning will be that a set is an element of The if and only in this case it is also an element of B. In other words, both sets have exactly the same elements.

It is very interesting that we do not know yet if there is any set. The first truth of ZF theory It tells us nothing about whether the universe of sets is populated by any set or not. The first truth ZF-1 it only clarifies what it means to say that one set is the same as another. Seems little, doesn't it? But it is extremely important that we have a precise meaning of the terms and phrases we write, as well as the ideas we think and express through symbols. You know this is of fundamental importance, we don't need to go into more detail. If someone still does not understand this, just buy a newspaper at the first newsstand they can find, read it carefully and try to find out the amount of "truths" in it. Anyway, our reflection does not admit “winding”, “lies”, “double meaning”Or“multiple directions”, Or any other kind of“verbal maneuver”Or“writing”To be“deceived" It is a great happiness that we can “experiment mathematically”The possibility of building“a world of reliable truths”.

We will not be immune to mistakes, that is, we may come to “fool ourselves into crafting truths" So it is possible for us to make a mistake and thus it is possible for us to say, “lies" But we have the strong belief that once a “mistake”, “a mistake” or “a lie” has been pointed out by someone who has a “logical argument”, Everyone else is convinced of the“ mistake ”and the episode is clarified forever. For example, at some point we will demonstrate that there are numbers. Everyone will be convinced of our demonstration, or someone will point out a mistake that will be unequivocally accepted by everyone. Thus, we do not believe in the possibility of "winding" against each other in order to obtain unquestionable favors. If our view is acceptable a conclusion is inevitable: in our country it would be a revolution, perhaps the greatest in all of our history, if a few million citizens learned to elaborate truths mathematically and thus to perceive "deceptions". Let's proceed with one challenge for you: explain in your words, and give an example of it, the second truth of Zermelo-Fraenkel set theory:

ZF (2) Axiom Scheme for Sub-Assembly Formation: for all The, exist Bsuch that: for all x, x belongs to B if and only if x belongs to The and worth the property THE(x).

Economically (in mathematical symbols):

Good luck and see you next time we'll take a closer look at this second mathematical truth.

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