Abstract
Solomon Feferman has left his mark on computability theory, as on many other areas of foundational studies. The purpose of this paper is, by means of reviewing a selected few of his many papers in this area, to give an idea of his impressive insights and developments in this field.
To the memory of Sol Feferman, who, with his unfailing kindness, patience and good humour, was a constant source of inspiration to me, ever since his supervision of my graduate studies at Stanford
J. Zucker—Thanks to Wilfried Sieg and an anonymous referee for very helpful comments on an earlier draft, and to Mark Armstrong for technical assistance with the typesetting. This research was supported by a grant from the Natural Sciences and Engineering Research Council (Canada).
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Notes
- 1.
- 2.
And regrettably never published, but with a far-ranging influence, as we will see in some of the other papers investigated here.
- 3.
This terminology is SF’s.
- 4.
- 5.
Where ‘Sc’ is the successor function on \(\mathbb N\).
- 6.
Gandy to SF, personal communication.
- 7.
- 8.
- 9.
Actually a congruence relation w.r.t. the \(F_k\)’s.
- 10.
Emphasis added. The meaning of this phrase is discussed below.
- 11.
As noted in §1 above.
- 12.
Below I use ‘\(\mathbb N\)’ for ‘\(\omega \)’.
- 13.
- 14.
Emphasis added. This gives the essential difference between models 2 and 3.
- 15.
Emphasis added. This gives the essential difference between models 2 and 3.
- 16.
Recall (2.1) the equality relations ‘\(=_{A_i}\)’ on \(\mathcal A\).
- 17.
- 18.
That is, model 4 described above.
- 19.
- 20.
Previously reviewed by me in [60], which lists some (minor) slips in this paper.
- 21.
As in (3.1).
- 22.
My terminology.
- 23.
My terminology.
- 24.
Or more accurately, simultaneously with this lemma.
- 25.
As SF points out, this implies that \(F\!\upharpoonright \!B:B_{\bar{\sigma }}\times B_{\bar{\imath }}\overset{\sim }{\rightarrow }B_j\), but not conversely.
- 26.
- 27.
Note that this is the stream structure (2.5) without the equalities.
- 28.
- 29.
I.e., any linearly ordered subset of \(A_i\) has a l.u.b in \(A_i\).
- 30.
Not given explicitly in [18].
- 31.
As we will see with the recursion scheme shown below.
- 32.
In connection with another recursion scheme, but it is still appropriate here.
- 33.
- 34.
Unfortunately never published (personal communication by SF).
- 35.
See e.g. [56] and the references therein.
- 36.
Discussed further in §4 below.
- 37.
- 38.
Emphasis added. Following SF, I shall focus on the real-computability aspect of the various models, rather than their complexity-theoretical or scientific-computational aspects.
- 39.
Emphasis added.
- 40.
See footnote 4.
- 41.
We use “algebra” rather than “structure” to indicate that their signatures contain function (and constant) but not relation symbols.
- 42.
See footnote 4.
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Zucker, J. (2017). Feferman on Computability. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_2
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