Introduction
History
A case study
Conclusion
The original text of my slides is in black. I have added some
comments, corrections and clarifications based on the (very
interesting) discussion which followed my talk. This extra
material is in red.
1 / 19
Introduction
History
A case study
Conclusion
Cardinal arithmetic and set-theoretic
methodology
James Cummings
CMU
17 October 2010
www.math.cmu.edu/users/jcumming/phil/slides.pdf
2 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
3 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
(Shelah) If A is a progressive interval of regular cardinals then
|pcf (A)| 6 |A|+3 .
3 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
(Shelah) If A is a progressive interval of regular cardinals then
|pcf (A)| 6 |A|+3 .
(Gitik) The consistency strength of the failure of the Singular
Cardinals Hypothesis is exactly o(κ) = κ++ .
3 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
(Shelah) If A is a progressive interval of regular cardinals then
|pcf (A)| 6 |A|+3 .
(Gitik) The consistency strength of the failure of the Singular
Cardinals Hypothesis is exactly o(κ) = κ++ .
(Harrington-Kechris-Louveau) A Borel equivalence relation on
a Polish space is either smooth or continuously embeds E0 .
3 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
(Shelah) If A is a progressive interval of regular cardinals then
|pcf (A)| 6 |A|+3 .
(Gitik) The consistency strength of the failure of the Singular
Cardinals Hypothesis is exactly o(κ) = κ++ .
(Harrington-Kechris-Louveau) A Borel equivalence relation on
a Polish space is either smooth or continuously embeds E0 .
(Viale) The Proper Forcing Axiom implies the Singular
Cardinals Hypothesis.
3 / 19
Introduction
History
A case study
Conclusion
Some notable results in cardinal arithmetic (broadly construed)
from the last 25 years or so:
(Shelah) If A is a progressive interval of regular cardinals then
|pcf (A)| 6 |A|+3 .
(Gitik) The consistency strength of the failure of the Singular
Cardinals Hypothesis is exactly o(κ) = κ++ .
(Harrington-Kechris-Louveau) A Borel equivalence relation on
a Polish space is either smooth or continuously embeds E0 .
(Viale) The Proper Forcing Axiom implies the Singular
Cardinals Hypothesis.
(Woodin) If the Ω-conjecture holds and the theory of H(ω2 ) is
finitely axiomatised over ZFC in Ω-logic then CH fails.
3 / 19
Introduction
History
A case study
Conclusion
Things we might notice about these results:
4 / 19
Introduction
History
A case study
Conclusion
Things we might notice about these results:
1
Logic plays a prominent role. Gitik’s result is a theorem
about theories, Viale’s result involves a new axiom,
Woodin’s result involves a new logic.
4 / 19
Introduction
History
A case study
Conclusion
Things we might notice about these results:
1
Logic plays a prominent role. Gitik’s result is a theorem
about theories, Viale’s result involves a new axiom,
Woodin’s result involves a new logic.
2
Many of them involve concepts (cardinals of high Mitchell
order, proper forcing posets, cofinalities of ultraproducts)
which are historically quite recent and not obviously
related to cardinal arithmetic.
4 / 19
Introduction
History
A case study
Conclusion
Things we might notice about these results:
1
Logic plays a prominent role. Gitik’s result is a theorem
about theories, Viale’s result involves a new axiom,
Woodin’s result involves a new logic.
2
Many of them involve concepts (cardinals of high Mitchell
order, proper forcing posets, cofinalities of ultraproducts)
which are historically quite recent and not obviously
related to cardinal arithmetic.
3
Shelah’s result is the most “classical”, in the sense that it is
a combinatorial theorem proved in ZFC. However neither
the hypotheses nor the conclusion explicitly mention
cardinal arithmetic.
4 / 19
Introduction
History
A case study
Conclusion
Things we might notice about these results:
1
Logic plays a prominent role. Gitik’s result is a theorem
about theories, Viale’s result involves a new axiom,
Woodin’s result involves a new logic.
2
Many of them involve concepts (cardinals of high Mitchell
order, proper forcing posets, cofinalities of ultraproducts)
which are historically quite recent and not obviously
related to cardinal arithmetic.
3
Shelah’s result is the most “classical”, in the sense that it is
a combinatorial theorem proved in ZFC. However neither
the hypotheses nor the conclusion explicitly mention
cardinal arithmetic.
4
The Harrington-Kechris-Louveau theorem is about
“effective cardinal arithmetic”, and involves subtle
distinctions between sets of size 2ℵ0 .
4 / 19
Introduction
History
A case study
Conclusion
Roughly a century has passed since Hausdorff (1908)
enunciated the Generalised Continuum Hypothesis and
introduced weakly inaccessible cardinals. Rather arbitrarily I’ll
take 1908 as the start of the modern era in cardinal arithmetic.
5 / 19
Introduction
History
A case study
Conclusion
Roughly a century has passed since Hausdorff (1908)
enunciated the Generalised Continuum Hypothesis and
introduced weakly inaccessible cardinals. Rather arbitrarily I’ll
take 1908 as the start of the modern era in cardinal arithmetic.
During the modern era there have been radical transformations
in our conception of the objects and methods in this area. This
poses (or at least it should pose) a challenge for the philosophy
of set theory.
5 / 19
Introduction
History
A case study
Conclusion
A piece of received wisdom: Since cardinal addition and
multiplication is trivial, the only interesting thing in cardinal
arithmetic is exponentiation.
6 / 19
Introduction
History
A case study
Conclusion
A piece of received wisdom: Since cardinal addition and
multiplication is trivial, the only interesting thing in cardinal
arithmetic is exponentiation.
“We think the real problems are on pcf, whereas cardinal
arithmetic problems are “artificial” remnants of them which the
“noise” of 2λ (λ regular) does not drown. So the real problem is
0
“can |pcf (a)| > |a|?” while the “artificial remnant” is “can ℵℵ
ω
ℵ
be > ℵω1 + 2 0 ?”
Saharon Shelah, from the Introduction to “Cardinal arithmetic”
6 / 19
Introduction
History
A case study
Conclusion
Some natural questions:
7 / 19
Introduction
History
A case study
Conclusion
Some natural questions:
1
How did we get here?
7 / 19
Introduction
History
A case study
Conclusion
Some natural questions:
1
How did we get here?
2
What has become of “cardinal arithmetic” as a subject?
Has it just disintegrated?
7 / 19
Introduction
History
A case study
Conclusion
Some natural questions:
1
How did we get here?
2
What has become of “cardinal arithmetic” as a subject?
Has it just disintegrated?
3
Using cardinal arithmetic as a case study, can we draw any
conclusions about the philosophy of set theory?
7 / 19
Introduction
History
A case study
Conclusion
In the rest of this talk I will:
8 / 19
Introduction
History
A case study
Conclusion
In the rest of this talk I will:
Discuss the history of cardinal arithmetic.
8 / 19
Introduction
History
A case study
Conclusion
In the rest of this talk I will:
Discuss the history of cardinal arithmetic.
Look in some detail at a modern result.
8 / 19
Introduction
History
A case study
Conclusion
In the rest of this talk I will:
Discuss the history of cardinal arithmetic.
Look in some detail at a modern result.
Suggest some challenges for philosophers of set theory.
8 / 19
Introduction
History
A case study
Conclusion
“What is discussed is the tendency in many historians . . . to
praise revolutions provided they have been successful, to
emphasize certain principles of progress in the past and to
produce a story which is the ratification if not the glorification
of the present.”
Herbert Butterfield, from the Preface to The Whig Interpretation
of History (1931)
What follows is an extremely incomplete, biased and Whiggish
list of some important results in cardinal arithmetic between
about 1908 and 1980. The main intent is to provide some
historical background for the result by Shelah discussed in
Section Three, “A case study”. The dates given are dates of
publication, which in many cases do not accurately reflect
when the work was done.
9 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
Gale and Stewart (1953): Open determinacy.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
Gale and Stewart (1953): Open determinacy.
Hajnal, Lévy (1956, 1957) Relative constructibility.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
Gale and Stewart (1953): Open determinacy.
Hajnal, Lévy (1956, 1957) Relative constructibility.
Scott(1961): Ultrapowers of V: GCH does not fail first at a
measurable cardinal.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
Gale and Stewart (1953): Open determinacy.
Hajnal, Lévy (1956, 1957) Relative constructibility.
Scott(1961): Ultrapowers of V: GCH does not fail first at a
measurable cardinal.
Cohen (1963): Forcing: independence of AC, GCH.
10 / 19
Introduction
History
A case study
Conclusion
Hausdorff (1908): Formulation of GCH, inaccessible cardinals.
Ulam (1930): Measurable cardinals.
Gödel (1931): Incompleteness theorem.
Gödel (1940): The constructible universe: relative consistency
of AC, GCH.
Gale and Stewart (1953): Open determinacy.
Hajnal, Lévy (1956, 1957) Relative constructibility.
Scott(1961): Ultrapowers of V: GCH does not fail first at a
measurable cardinal.
Cohen (1963): Forcing: independence of AC, GCH.
Easton(1970): The continuum function can behave with
complete freedom on regular cardinals: essentially the only
constraints are κ < λ =⇒ 2κ 6 2λ and cf (2κ ) > κ.
10 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
Solovay(1971): The SCH holds above a strongly compact
cardinal.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
Solovay(1971): The SCH holds above a strongly compact
cardinal.
Silver (1974): The SCH does not first fail at a singular cardinal
of uncountable cofinality.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
Solovay(1971): The SCH holds above a strongly compact
cardinal.
Silver (1974): The SCH does not first fail at a singular cardinal
of uncountable cofinality.
Jensen (1974): The Covering Lemma for L. 0] as lower bound
for failure of SCH.
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
Solovay(1971): The SCH holds above a strongly compact
cardinal.
Silver (1974): The SCH does not first fail at a singular cardinal
of uncountable cofinality.
Jensen (1974): The Covering Lemma for L. 0] as lower bound
for failure of SCH.
Magidor (1977) SCH can fail at ℵω . GCH can first fail at ℵω .
11 / 19
Introduction
History
A case study
Conclusion
Kunen (1970): L[µ], iterated ultrapowers.
Martin (1971): Sharps imply analytic determinacy.
Prikry, Silver (197?) The Singular Cardinals Hypothesis (SCH)
can fail from large cardinals.
Solovay(1971): The SCH holds above a strongly compact
cardinal.
Silver (1974): The SCH does not first fail at a singular cardinal
of uncountable cofinality.
Jensen (1974): The Covering Lemma for L. 0] as lower bound
for failure of SCH.
Magidor (1977) SCH can fail at ℵω . GCH can first fail at ℵω .
Dodd, Jensen (1981, 1982) The core model KDJ for one
measurable cardinal, covering lemmas for L[µ] and KDJ .
11 / 19
Introduction
History
A case study
Conclusion
Comments on the history: Tony Martin observed that the
history of modern work on determinacy should be traced back
to the Banach-Mazur game (mid-1930s). Aki Kanamori
observed in conversation that Silver’s theorem on the least
failure of GCH was motivated by some earlier work of
Magidor, who got similar conclusions with an extra hypothesis.
12 / 19
Introduction
History
A case study
Conclusion
1
(Shelah 1986) If ℵδ is the ω1 -th cardinal fixed point then ℵℵ
δ is
ℵ1
less than the (22 )+ -th cardinal fixed point.
13 / 19
Introduction
History
A case study
Conclusion
1
(Shelah 1986) If ℵδ is the ω1 -th cardinal fixed point then ℵℵ
δ is
ℵ1
less than the (22 )+ -th cardinal fixed point.
Main idea of the proof: Either there are many inner models
with Ramsey cardinals, or there are not.
13 / 19
Introduction
History
A case study
Conclusion
1
(Shelah 1986) If ℵδ is the ω1 -th cardinal fixed point then ℵℵ
δ is
ℵ1
less than the (22 )+ -th cardinal fixed point.
Main idea of the proof: Either there are many inner models
with Ramsey cardinals, or there are not.
CASE ONE: There are not many inner models with Ramsey
cardinals. In this case we can appeal to instances of Jensen
covering over inner models with enough GCH.
13 / 19
Introduction
History
A case study
Conclusion
1
(Shelah 1986) If ℵδ is the ω1 -th cardinal fixed point then ℵℵ
δ is
ℵ1
less than the (22 )+ -th cardinal fixed point.
Main idea of the proof: Either there are many inner models
with Ramsey cardinals, or there are not.
CASE ONE: There are not many inner models with Ramsey
cardinals. In this case we can appeal to instances of Jensen
covering over inner models with enough GCH.
CASE TWO: There are many inner models with Ramsey
cardinals. In this case we can (via an appeal to a form of
analytic determinacy) build an ideal on ℵ1 with a weak form of
precipitousness, then use the ideal to get the bound.
13 / 19
Introduction
History
A case study
Conclusion
Shelah has revisited this result several times. The relevant
sources are (Sh:75) “A note on cardinal exponentiation” JSL 45
(1980) 55–66, (Sh:111) “On power of singular cardinals” NDJFL
27 (1986) 263–299, (Sh:256) “More on powers of singular
cardinals” IJM 59 (1987) 299-326, and Chapter V of “Cardinal
Arithmetic”.
14 / 19
Introduction
History
A case study
Conclusion
Ingredients:
15 / 19
Introduction
History
A case study
Conclusion
Ingredients:
Large cardinals.
15 / 19
Introduction
History
A case study
Conclusion
Ingredients:
Large cardinals.
Forcing.
15 / 19
Introduction
History
A case study
Conclusion
Ingredients:
Large cardinals.
Forcing.
Infinite games, determinacy of games from large cardinals.
15 / 19
Introduction
History
A case study
Conclusion
Ingredients:
Large cardinals.
Forcing.
Infinite games, determinacy of games from large cardinals.
Inner models, sharps, and covering lemmas.
15 / 19
Introduction
History
A case study
Conclusion
Ingredients:
Large cardinals.
Forcing.
Infinite games, determinacy of games from large cardinals.
Inner models, sharps, and covering lemmas.
Generic elementary embeddings (precipitousness).
15 / 19
Introduction
History
A case study
Conclusion
Challenge for philosophers: say something concrete about
16 / 19
Introduction
History
A case study
Conclusion
Challenge for philosophers: say something concrete about
the objects of set theory,
16 / 19
Introduction
History
A case study
Conclusion
Challenge for philosophers: say something concrete about
the objects of set theory,
the methods of set theory,
16 / 19
Introduction
History
A case study
Conclusion
Challenge for philosophers: say something concrete about
the objects of set theory,
the methods of set theory,
or the status of set-theoretic claims,
16 / 19
Introduction
History
A case study
Conclusion
Challenge for philosophers: say something concrete about
the objects of set theory,
the methods of set theory,
or the status of set-theoretic claims,
which (if taken seriously by practitioners of set theory) would
not have impeded the developments necessary for the proof of
Shelah’s result.
On reflection this seems like an unduly negative formulation. I
don’t want to suggest that a philosophical stance is always a
barrier to progress.
16 / 19
Introduction
History
A case study
Conclusion
Joel Hamkins asked for an example of a philosophical stance
that would have obstructed progress. I gave the example of
“mathematics does not need new axioms”.
Juliette Kennedy took issue with the term concrete, saying that
by the time a philosopher knows enough set theory to say
something concrete they have become a set theorist. I don’t
disagree but I don’t see this as a problem.
17 / 19
Introduction
History
A case study
Conclusion
18 / 19
Introduction
History
A case study
Conclusion
“The following essay is written in the conviction that anarchism,
while perhaps not the most attractive political philosophy, is
certainly excellent medicine for epistemology and for the
philosophy of science . . . it will become clear that there is only one
principle that can be defended under all circumstances and in
all stages of human development. It is the principle: anything
goes.
Paul Feyerabend, “Against method” (1988)
18 / 19
Introduction
History
A case study
Conclusion
In response to comments by Joel Hamkins, Kai Hauser, and
Peter Koellner, I will try to clarify my position:
I am not advocating that everyone should have the same
“anarchist” philosophical stance (I am not even sure what such
a thing would be) nor that everyone should have no
philosophical stance (I am not even sure this is possible). I
think that it is healthy for many philosophical stances to exist
and contend.
I think that philosophy of mathematics should be engaged with
the actual practices of mathematicians, and respectful of the
autonomy of mathematics. I also think that set theory is part of
mathematics.
19 / 19