Red workbook, p10

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.