# The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

is good to 0. If Ϛ is good to n, then Ϛ ⋃ {< n, least member of α not in the range of Ϛo>} is good to sn (the existence of the least member of α not in the range of Ϛ follows from our supposition). By induction, for every n, some o is good to n. If Ϛ is good to n, Ϛ′ is good to n′, and n≤n′, then by induction for every ii) = Ϛ'(i). By comprehension, let ρ = {〈n, k〉: ∃Ϛ (Ϛ is good to sn and Ϛ(n) = k)}. It is sufficiently clear that ρ is a function; domain (ρ) = N; for all i, ρ(i) ∈ α; if ji), and if k ∈ α, k < ρ(i), then for some ji) whenever j

APPENDIX 3. TRANSLATIONS INTO
PRESENT-DAY NOTATION OF §§115, 117, 119 AND
PART OF §§113 OF GRUNDGESETZE

Below we use boldface instead of Fraktur and notation introduced above in place of Frege's own, we omit the signs indicating which rule of inference is applied and reference numbers to axioms of Frege's system, and we utilize certain easy equivalences (e.g. p ∧ q for ¬(p → ¬q)).

§113 (part)

aq*b → ∃e(eqb ∧ aq*=e) (141)

∀e(eqb → ¬q*=e) → ¬q*b (142)

¬aP*=d → d=c → ¬aP*=c

(88): ¬aP*=d → dPb → cPb → ¬aP*=c

¬aP*=d → dPb → ∀e(ePb → ¬aP*=e)

(142): ¬aP*=d → dPb → ¬aP*b

aP*b → dPb → aP*=d (143)

§115
130 bq*=a → ¬bq*a → a=b

¬bq*a → bq*=a → b=a (146)

bq*=a ∧ b≠a → bq*a (147)

147 bP*=a ∧ b≠a → bP*a

(143): dPa → (bP*=a A b≠a) → bP*=d

(bP*=d → (bP*=a ∧ b≠a)) → dPa → ((bP*=a ∧ b≠a) ↔ bP*=d) (β)

134 bP*=d → dPa → bP*a
¬,aP*a → bP*=d → dPa → b≠a

(145): OP*=a → bP*"d → dPa → b≠a

bP*=a → OP*=a → bP*=d → dPa → (bP*=a ∧ b≠a)

(137): dPa → OP*=a → bP*=d → (bP*=a ∧ b≠a)

(β): dPa → OP*=a → ((bP*=a ∧ b≠a) ↔ bP*=d)

(77): dPa → OP*=a → ([x: xP*=a a x≠a]b ↔ bP*=d) (148)

dPa → OP*=a → ∀a([x: xP*=a A x≠a]a ↔ aP*=d)

(96): dPa → OP*=a → #[x: xP*=a ∧ x≠a] = #[x: xP*=d] (149)

-426-

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes

#### Cited page

Style
Citations are available only to our active members.
Upgrade your membership to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

#### Cited page

The Philosophy of Mathematics Today

Settings

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.

Why can't I print more than one page at a time?

Help
Full screen
Items saved in your active project from this book
• Bookmarks
• Highlights & Notes
• Citations
/ 646

### How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

## Cited passage

Style
Citations are available only to our active members.
Upgrade your membership to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Cited passage

Search by...
Show...

### Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.