# 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.