# A method for proving something about all sets in ZFC

• phoenixthoth
In summary, the author discusses an induction principle that is applicable to grammatical systems, and a result that is known to be true in set theory.
phoenixthoth
I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in ZFC set theory (and perhaps other set theories) which would enable someone to prove statements about all sets in ZFC.

So what kind of feedback do you want? Everything in the document is rather well-known, it's only phrased in terms that are pretty weird. There are some inaccurate things in the document though, so perhaps you want to discuss that?

In particular, take a look at section 1.2 of Hinman's "Fundamentals of logic". You should be more careful in your document however since you only define "grammatical systems" for sets, while you later use them for classes when dealing with ZF.

micromass said:
So what kind of feedback do you want? Everything in the document is rather well-known, it's only phrased in terms that are pretty weird. There are some inaccurate things in the document though, so perhaps you want to discuss that?
That in itself is very useful feedback. Thank you. I guess I haven't studied enough set theory because I haven't seen this result before. I'm glad I don't have to reinvent the wheel. I will see if I can find Hinman. We might as well discuss those innacurate things. I don't know if changing the terms I use from sets to classes when defining grammatical systems would harm anything. I was also thinking, alternatively, that maybe I should note that I must not be working within ZFC in order to prove something about ZFC; I apparently need something else like NFU or something that isn't ZFC. I'm excited to go find out other presentations of this result which I was suspecting was well-known, in part to see if more or less machinery is required in other presentations.

Are there any references online that explain this result differently from how I did it (or almost did it) as I cannot find Hinman's book online?

## 1. What is ZFC?

ZFC stands for Zermelo-Fraenkel set theory with the axiom of choice. It is a commonly used axiomatic system in set theory that is used to prove mathematical statements about sets.

## 2. What is the method for proving something about all sets in ZFC?

The method for proving something about all sets in ZFC is to use the axioms and rules of ZFC to construct a logical proof. This typically involves defining sets and using logic to show that the desired statement holds for all sets.

## 3. Why is it important to prove something about all sets in ZFC?

Proving something about all sets in ZFC is important because it allows us to make general statements that hold for all sets, rather than just specific examples. This can help us understand the properties and behavior of sets in a broader context.

## 4. What kind of statements can be proven about all sets in ZFC?

Many different types of statements can be proven about all sets in ZFC, including statements about set equality, cardinality, and relationships between sets. However, not all statements can be proven using ZFC, as there are some well-known mathematical statements, such as the continuum hypothesis, that are independent of ZFC.

## 5. Are there other axiomatic systems that can be used to prove statements about sets?

Yes, there are other axiomatic systems that can be used to prove statements about sets, such as the von Neumann-Bernays-Gödel set theory and Morse-Kelley set theory. These systems have different sets of axioms and rules, and may be better suited for proving certain types of statements about sets.

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