Red workbook, p10

Source

Transcript

• Notiz:
  • wieso \mathbb{Z} + p \subseteq \mathbb{N^* } fuer \mathbb{N^* } \ni p (evtl. min. id.pot.)
  • Kopie von \mathbb{Z}?
  • [this note was struck out by a check mark]
• Frage nicht standard PA-Modelle:
  • gibt es Eigenschaften, die sich reflektieren lassen?
  • Gibt es Reflektionsprinzip?
  • [the above two lines were struck out with as single line]
  • [illegible] Wenn es ein [struck out]
  • Idee: Wenn ein Nicht standard element
  • eine Eigenschaft hat
  • so haben unendlich viele Numerale
  • diese Eigenschsft, also
  • unendlich viele Nicht-Standard
    • [da elementare Substr]
  • Richtig?
  • Was kann man damit machen?

partial Translation

• Note:
  • Why is \mathbb{Z} + p \subseteq \mathbb{N^* } for any \mathbb{N^* } \ni p (possibly just minimal idempotent)
  • Copy of \mathbb{Z}?
  • [this note was struck out by a check mark]
• Question about non-standard models of PA
  • are there properties that can be reflected?
  • Are there reflection principles?
  • [the above two lines were struck out with as single line]
  • Idea: If a non-standard element has some property, then infinitely many numerals have this property and therefore infinitely many non-standard elements have this property (as an elementary substructure).
  • Correct? What can you do with it?

Notes

The previous page is followed by another attempt of research ideas.

First there's a note on a basic but important observation for \beta \mathbb{N} -- it contains lots of copies of \mathbb{Z}. I remember trying to figure this out and ending up asking Sabine Koppelberg -- and the solution took two second, leaving me miserably disappointed by my failure.

To understand the second part of the note, I should explain that my Diplom thesis was about large cardinals and reflection principles. The first few things I tried that summer came out of that perspective -- looking at cardinals (as a semingroup with ordinal addition/multiplication), hoping to connect with large cardinal theory. Nothing ever came of it but perhaps we'll encounter that later in this workbook.