Proof of Hartog's Theorem: Axiom of Replacement Applied

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Discussion Overview

The discussion revolves around the proof of Hartog's Theorem, specifically focusing on the application of the Axiom of Replacement within the proof. Participants explore the steps involved in the proof, particularly how to highlight the role of the Axiom of Replacement and the construction of first-order formulas.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant requests a careful rewriting of the proof of Hartog's Theorem, emphasizing the use of the Axiom of Replacement.
  • Another participant explains that the proof involves demonstrating that the class of all order-types of well-orders is a set, which utilizes the Axiom of Replacement through a first-order formula expressing isomorphism.
  • A question is raised about the construction of a first-order formula, indicating a need for clarification on this aspect.
  • A later reply assumes familiarity with first-order logic among participants studying formal Set Theory, suggesting that understanding this logic is essential for grasping the proof.

Areas of Agreement / Disagreement

Participants appear to agree on the importance of the Axiom of Replacement in the proof, but there is uncertainty regarding the construction of first-order formulas and the level of familiarity with first-order logic required for understanding the proof.

Contextual Notes

The discussion does not resolve the specifics of constructing first-order formulas or the implications of the Axiom of Replacement in detail, leaving these aspects open for further exploration.

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Write out the proof of Hartog's Theorem again carefully highlighting how the Axiom of Replacement is used

How can you highlight the axiom of replacment?
 
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In the proof of Hartog's theorem, you reach a point where you just proved that the class of all well-orders of the set X (using the separation axiom schema). Now you have to prove that each the class of all order-types (isomorphic ordinals) of each of these well-orders is also a set, and it's here that you use replacement, by constructing a first-order formula that expresses the isomorphism of each (Y,<), where Y is a subset of X and "<" is a well-order in Y, to an ordinal.
This is the key step in Hartog's proof, the one that allows him to sidestep AC, so I think the "highlighting" is a careful, step-by-step explanation of how the replacement schema is used.
 
That makes sense, but how do you construct a first order formula?
 
I assume that, if you're studying formal Set Theory, that you are familiar with first-order logic (otherwise, it's impossible to understand anything of Set Theory).
 

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