Peter Krautzberger on the web

Red workbook, p10


red workbook, p10
Red Workbook, p.10


  • 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?


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.