A method for proving something about all sets in ZFC

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Discussion Overview

The discussion revolves around a document proposing a method for generalizing induction in the context of ZFC set theory, particularly through the concept of "grammatical systems." Participants are providing feedback on the document's content, accuracy, and the implications of the proposed method.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks feedback on their document that generalizes induction in ZFC set theory through grammatical systems.
  • Another participant questions the originality of the ideas presented, suggesting that they are well-known but expressed in unusual terms.
  • Concerns are raised about inaccuracies in the document, particularly regarding the definition of "grammatical systems" and their application to classes versus sets.
  • A participant expresses uncertainty about whether changing terminology from sets to classes would affect the validity of their arguments.
  • There is a suggestion that the author may need to work outside of ZFC to prove statements about ZFC, possibly requiring alternative set theories like NFU.
  • The author expresses a desire to find other presentations of the result to understand the necessary machinery involved.
  • A request is made for online references that explain the result differently from the author's approach, indicating difficulty in accessing Hinman's book.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the accuracy and originality of the document's content, with multiple views on its validity and the definitions used. The discussion remains unresolved regarding the implications of the proposed method and the necessary theoretical framework.

Contextual Notes

Limitations include potential inaccuracies in the document, the unclear relationship between sets and classes in the context of grammatical systems, and the need for further exploration of alternative set theories.

phoenixthoth
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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.

https://docs.google.com/document/d/1amDb4Yti4egpKfcO2oLcnGAH8UpC8_tKb7ivuH3AT7A/edit?usp=sharing
 
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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?
 

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