This question arises out of having Devlin's Constructibility [1] in my collection of books at home during the lockdown. Chapter IX of the book deals with Silver machines, which are presented as Silver's attempt at capturing the fine structural content of $L$ using a coherent hierarchy of functions.

Searching for "Silver machine" on MathSciNet yields four results, one of which is Devlin's book and the other three which are currently inaccessible to me on the internet.

Devlin uses Silver machines to prove a global $\square$ principle in $L$, and has an exercise that asks the reader to prove the Covering Lemma(!) using them as well. It appears that this may be done in one of the four references from MathSciNet, and the title of another advertises the construction of a gap-1 morass using Silver's ideas. This shows that such "machines" should have a lot of power, but they also seem to have not attracted the attention of set-theorists.

At an abstract level, I am wondering "Why is this?" and even "What are Silver machines good for?"

For a more concrete question:

Are there significant consequences of V=L for which there is no known "Silver machine proof"? In particular, can one establish the existence of gap-n morasses using Silver machines?

Any references would be appreciated. Devlin's discussion mentions unpublished work of Silver, and also unpublished notes of Litman.

[1] Devlin, Keith J., Constructibility, Perspectives in Mathematical Logic. Berlin etc.: Springer-Verlag. XI, 425 p. DM 158.00; {$} 57.60 (1984). ZBL0542.03029.

  • 1
    $\begingroup$ This is not my bread and butter, so excuse this comment if it is off. Looking at the wikipedia page, it seems like these are just abstracting some properties of "most" $L_\alpha$'s (with $\Sigma_1$-Skolem functions in place of the the $h_i$'s?). So I wonder, is it just a reformulation of basics, like $L_\alpha$ vs. $J_\alpha$? Is there is some kind of translation between "standard" proofs and Silver machine proofs? I don't see the combinatorial mojo they bring to the table. $\endgroup$ Apr 21, 2020 at 7:38
  • 2
    $\begingroup$ I know that silver has a better conductivity than copper, but copper has better antibacterial properties. In this day and age, did you consider using copper machines? $\endgroup$
    – Asaf Karagila
    Apr 21, 2020 at 10:49
  • $\begingroup$ That may be the answer to the question @MonroeEskew, and that they only reformulate the salient properties of the hierarchy. $\endgroup$ Apr 21, 2020 at 13:35

2 Answers 2


The existence of morasses using Silver machine in L is proved in the PhD thesis of Thomas Lloyd Richardson, ``Silver Machine Approach to the Constructible Universe''.

See also Silver machines and Singular cardinals and analytic games. In this last paper the author relativises the Silver machine concept to an arbitrary $L[a], a⊆ω$, and using this relativised notion he establishes a relativised version of the Jensen covering lemma.


To begin to answer your 'abstract question', "What are Silver Machines good for', one need only look at the title of Prof. Silver's unpublished manuscript

"How to eliminate the fine structure from the work of jensen"

to find the beginning of the answer.

Why does one wish to eliminate the fine structure of Jensen? For the reason given by Boris Piwinger in section 0 of his Diploma Thesis found by clicking on the link "Silver Machines" found in Prof. Golshani's answer (I am quoting from an English translation I found on the Web--on pp. 2-3 of the translation):

The idea [of fine structure theory--my comment] is to have a closer look at the passage from $L_{\alpha}$ to $L_{\alpha + 1}$ and to describe the process with some "bookkeeping device". Even today--after 25 years of development--, this method is extremely complicated and wearisome [is it really?--my question].

This is the correct context in which to frame the OP's question (at least in my opinion).

Addendum: I also find the following passage found on pg.3 of Piwinger's Thesis of interest and relevance here:

Also in the early seventies, Jack Silver found a different approach--the Silver machines. These machines reduce the considerations to calculations on sets of ordinals.

This leads me to wonder whether Silver machines can be considered as a type of Ordinal Turing Machine (OTM) and by the following theorem

A set $x$ of ordinals is ordinal computable form a finite set of ordinal parameters if and only if it is an element of the constructible universe $L$.

one can reduce the primitive recursive set functions of Jensen and the Silver machines to this seemingly irreducible form (OTM's). This would serve to eliminate (hopefully) the concern regarding the "reformulating the salient properties of the (Jensen) hierarchy" Prof. Eisworth has regarding the Silver machines (though reformulating to simplify is certainly an important gain) and perhaps provide a means of classifying the various Fine Structure Hierarches as particular classes of OTM's


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.