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AMS-ASL Summer Institute
in
Axiomatic Set Theory

OFFICIAL BALLOT

[pencilled in: "80 ballots cast"]

I. A. I believe that the proposition

$\ \ \ \ \ \ \ \ \ $'The axiom MC of measurable cardinals is true in the real universe of sets'

is

$\quad$(38) meaningful $\quad$ (38) meaningless

$\ \ \ \ $B. (To be answered only if your answer to A is "meaningful")

$\ \ \ \ \ \ \ \ \ $(8) I think that MC is almost certainly true
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely true than false
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely false than true
$\ \ \ \ \ \ \ \ \ $(2) I think MC is almost certainly false
$\ \ \ \ \ \ \ \ \ $(14) I have no idea whether MC is true or false

$\ \ \ \ $C. Regarding the prediction that MC will someday be refuted in ZF,

$\ \ \ \ \ \ \ \ \ $(0) I think this prediction is almost certainly true
$\ \ \ \ \ \ \ \ \ $(2) I think this prediction is more likely true than false
$\ \ \ \ \ \ \ \ \ $(16) I think this prediction is more likely false than true
$\ \ \ \ \ \ \ \ \ $(8) I think this prediction is almost certainly false
$\ \ \ \ \ \ \ \ $(4) I have no idea whether this prediction is true or false

II. A. I believe that the proposition

'The continuum hypothesis CH is true in the real universe of sets'

is

$\quad$(42) meaningful $\quad$ (35) meaningless

$\ \ $B. (To be answered only if your answer to IIA is 'meaningful')

$\ \ \ \ \ \ $(2) I think CH is almost certainly true
$\ \ \ \ \ \ $(2) I think CH is more likely true than false
$\ \ \ \ \ \ $(12) I think CH is more likely false than true
$\ \ \ \ \ \ $(14) I think CH is almost certainly false
$\ \ \ \ \ \ $(12) I have no idea whether CH is true or false.
$\ \ $B'. (To be answered only if your answer to IIA is 'meaningless')

$\ \ \ \ $(1) My position on IIA

$\quad\quad$(2) does$\quad$(33) does not

$\ \ \ \ \ \ $cast doubt in my own mind on the value of set theory.

$\ \ \ \ $(2) I am inclined to think that set theory based on the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is destined to play in the long-range future develop-
$\ \ \ \ \ \ \ \ \ $ment of mathematics a

$\ \ \ \ \ \ \ \ \ $(11) more important role than
$\ \ \ \ \ \ \ \ \ $(13) role of equal importance with
$\ \ \ \ \ \ \ \ \ $(11) less important role than

$\ \ \ \ \ \ \ \ \ $set theory based on the denial of the continuum hypothesis.

$\ \ $ C. Assuming that human mathematicians still exist then, I believe that
$\ \ \ \ \ \ \ $in 2067 the prevailing opinion among them will be that the continuum
$\ \ \ \ \ \ \ $problem:

$\ \ \ \ $(4) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the continuum hypothesis is
$\ \ \ \ \ \ \ \ \ $a consequence
$\ \ \ \ $(18) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the denial of the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is a consequence
$\ \ \ \ $(37) has been settled by the general acceptance of the belief that
$\ \ \ \ \ \ \ \ \ $there is no one true set theory and that the continuum hypothesis
$\ \ \ \ \ \ \ \ \ $simply holds in some theories and fails in others
$\ \ \ \ $(11) is still unsettled

III. A. I believe that there is an absolute sense in which every sentence of
$\ \ \ \ \ \ \ \ \ $first-order number theory based on addition, multiplication, and
$\ \ \ \ \ \ \ \ \ $exponentiation is either true or false.

$\quad$(54) yes $\quad$ (26) no

$\ \ \ \ $B. I believe that there is an absolute sense in which every $\underline{\text{universal}}$
$\ \ \ \ \ \ \ \ \ $sentence of first-order number theory based on addition, multiplication,
$\ \ \ \ \ \ \ \ \ $and exponentiation is either true or false.

$\quad$(62) yes $\quad$ (18) no

Please do not sign your ballot.

August 1, 1967
University of California, Los Angeles

AMS-ASL Summer Institute
in
Axiomatic Set Theory

OFFICIAL BALLOT

[pencilled in: "80 ballots cast"]

I. A. I believe that the proposition

$\ \ \ \ \ \ \ \ \ $'The axiom MC of measurable cardinals is true in the real universe of sets'

is

$\quad$(38) meaningful $\quad$ (38) meaningless

$\ \ \ \ $B. (To be answered only if your answer to A is "meaningful")

$\ \ \ \ \ \ \ \ \ $(8) I think that MC is almost certainly true
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely true than false
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely false than true
$\ \ \ \ \ \ \ \ \ $(2) I think MC is almost certainly false
$\ \ \ \ \ \ \ $(14) I have no idea whether MC is true or false

$\ \ \ \ $C. Regarding the prediction that MC will someday be refuted in ZF,

$\ \ \ \ \ \ \ \ \ $(0) I think this prediction is almost certainly true
$\ \ \ \ \ \ \ \ \ $(2) I think this prediction is more likely true than false
$\ \ \ \ \ \ \ $(16) I think this prediction is more likely false than true
$\ \ \ \ \ \ \ \ \ $(8) I think this prediction is almost certainly false
$\ \ \ \ \ \ \ \ \ $(4) I have no idea whether this prediction is true or false

II. A. I believe that the proposition

'The continuum hypothesis CH is true in the real universe of sets'

is

$\quad$(42) meaningful $\quad$ (35) meaningless

$\ \ $B. (To be answered only if your answer to IIA is 'meaningful')

$\ \ \ \ \ \ $(2) I think CH is almost certainly true
$\ \ \ \ \ \ $(2) I think CH is more likely true than false
$\ \ \ \ $(12) I think CH is more likely false than true
$\ \ \ \ $(14) I think CH is almost certainly false
$\ \ \ \ $(12) I have no idea whether CH is true or false.
 
$\ \ $B'. (To be answered only if your answer to IIA is 'meaningless')

$\ \ \ \ $(1) My position on IIA

$\quad\quad$(2) does$\quad$(33) does not

$\ \ \ \ \ \ $cast doubt in my own mind on the value of set theory.

$\ \ \ \ $(2) I am inclined to think that set theory based on the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is destined to play in the long-range future develop-
$\ \ \ \ \ \ \ \ \ $ment of mathematics a

$\ \ \ \ \ \ \ \ \ $(11) more important role than
$\ \ \ \ \ \ \ \ \ $(13) role of equal importance with
$\ \ \ \ \ \ \ \ \ $(11) less important role than

$\ \ \ \ \ \ \ \ \ $set theory based on the denial of the continuum hypothesis.

$\ \ $ C. Assuming that human mathematicians still exist then, I believe that
$\ \ \ \ \ \ \ $in 2067 the prevailing opinion among them will be that the continuum
$\ \ \ \ \ \ \ $problem:

$\ \ \ \ $(4) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the continuum hypothesis is
$\ \ \ \ \ \ \ \ \ $a consequence
$\ \ $(18) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the denial of the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is a consequence
$\ \ $(37) has been settled by the general acceptance of the belief that
$\ \ \ \ \ \ \ \ \ $there is no one true set theory and that the continuum hypothesis
$\ \ \ \ \ \ \ \ \ $simply holds in some theories and fails in others
$\ \ $(11) is still unsettled

III. A. I believe that there is an absolute sense in which every sentence of
$\ \ \ \ \ \ \ \ \ $first-order number theory based on addition, multiplication, and
$\ \ \ \ \ \ \ \ \ $exponentiation is either true or false.

$\quad$(54) yes $\quad$ (26) no

$\ \ \ \ $B. I believe that there is an absolute sense in which every $\underline{\text{universal}}$
$\ \ \ \ \ \ \ \ \ $sentence of first-order number theory based on addition, multiplication,
$\ \ \ \ \ \ \ \ \ $and exponentiation is either true or false.

$\quad$(62) yes $\quad$ (18) no

Please do not sign your ballot.

August 1, 1967
University of California, Los Angeles

Here is a historical answer of sorts. I'm looking at a copy of a spirit-duplicated questionnaire, dated August 1, 1967, which was circulated at the AMS-ASL 1967 Summer Institute in Axiomatic Set Theory. The notation "80 ballots cast" is pencilled in, rather sloppily. The tally of votes for each answer is inked in by someone with neat handwriting. (It would be interesting to know if this survey has been published somewhere.)

Here is a historical answer of sorts. I'm looking at a copy of a spirit-duplicated questionnaire, dated August 1, 1967, which was circulated at the AMS-ASL 1967 Summer Institute in Axiomatic Set Theory. The notation "80 ballots cast" is pencilled in, rather sloppily. The tally of votes for each answer is inked in by someone with neat handwriting. From the numbers, I surmise that IC was only answered by those who answered "meaningless" to IA.

It would be interesting to know if this survey has been published somewhere.

Here is a historical answer of sorts. I'm looking at a copy of a spirit-duplicated questionnaire, dated August 1, 1967, which was circulated at the AMS-ASL 1967 Summer Institute in Axiomatic Set Theory. The notation "80 ballots cast" is pencilled in, rather sloppily. The tally of votes for each answer is inked in by someone with neat handwriting. The questionnaire is in three parts. Part I is about measurable cardinals, and part III is about first-order number theory. I will quote part II, which is about the continuum hypothesis. (It would be interesting to know if this survey has been published somewhere.)

II. A. I believe that the proposition

'The continuum hypothesis CH is true in the real universe of sets'

is

$\quad$(42) meaningful $\quad$ (35) meaningless

$\ \ $B. (To be answered only if your answer to IIA is 'meaningful')

$\ \ \ \ \ \ $(2) I think CH is almost certainly true
$\ \ \ \ \ \ $(2) I think CH is more likely true than false
$\ \ \ \ \ \ $(12) I think CH is more likely false than true
$\ \ \ \ \ \ $(14) I think CH is almost certainly false
$\ \ \ \ \ \ $(12) I have no idea whether CH is true or false.
$\ \ $B'. (To be answered only if your answer to IIA is 'meaningless')

$\ \ \ \ $(1) My position on IIA

$\quad\quad$(2) does$\quad$(33) does not

$\ \ \ \ \ \ $cast doubt in my own mind on the value of set theory.

$\ \ \ \ $(2) I am inclined to think that set theory based on the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is destined to play in the long-range future develop-
$\ \ \ \ \ \ \ \ \ $ment of mathematics a

$\ \ \ \ \ \ \ \ \ $(11) more important role than
$\ \ \ \ \ \ \ \ \ $(13) role of equal importance with
$\ \ \ \ \ \ \ \ \ $(11) less important role than

$\ \ \ \ \ \ \ \ \ $set theory based on the denial of the continuum hypothesis.

$\ \ $ C. Assuming that human mathematicians still exist then, I believe that
$\ \ \ \ \ \ \ $in 2067 the prevailing opinion among them will be that the continuum
$\ \ \ \ \ \ \ $problem:

$\ \ \ \ $(4) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the continuum hypothesis is
$\ \ \ \ \ \ \ \ \ $a consequence
$\ \ \ \ $(18) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the denial of the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is a consequence
$\ \ \ \ $(37) has been settled by the general acceptance of the belief that
$\ \ \ \ \ \ \ \ \ $there is no one true set theory and that the continuum hypothesis
$\ \ \ \ \ \ \ \ \ $simply holds in some theories and fails in others
$\ \ \ \ $(11) is still unsettled

Here is a historical answer of sorts. I'm looking at a copy of a spirit-duplicated questionnaire, dated August 1, 1967, which was circulated at the AMS-ASL 1967 Summer Institute in Axiomatic Set Theory. The notation "80 ballots cast" is pencilled in, rather sloppily. The tally of votes for each answer is inked in by someone with neat handwriting. (It would be interesting to know if this survey has been published somewhere.)

AMS-ASL Summer Institute
in
Axiomatic Set Theory

OFFICIAL BALLOT

[pencilled in: "80 ballots cast"]

I. A. I believe that the proposition

$\ \ \ \ \ \ \ \ \ $'The axiom MC of measurable cardinals is true in the real universe of sets'

is

$\quad$(38) meaningful $\quad$ (38) meaningless

$\ \ \ \ $B. (To be answered only if your answer to A is "meaningful")

$\ \ \ \ \ \ \ \ \ $(8) I think that MC is almost certainly true
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely true than false
$\ \ \ \ \ \ \ \ \ $(7) I think MC is more likely false than true
$\ \ \ \ \ \ \ \ \ $(2) I think MC is almost certainly false
$\ \ \ \ \ \ \ \ \ $(14) I have no idea whether MC is true or false

$\ \ \ \ $C. Regarding the prediction that MC will someday be refuted in ZF,

$\ \ \ \ \ \ \ \ \ $(0) I think this prediction is almost certainly true
$\ \ \ \ \ \ \ \ \ $(2) I think this prediction is more likely true than false
$\ \ \ \ \ \ \ \ \ $(16) I think this prediction is more likely false than true
$\ \ \ \ \ \ \ \ \ $(8) I think this prediction is almost certainly false
$\ \ \ \ \ \ \ \ $(4) I have no idea whether this prediction is true or false

II. A. I believe that the proposition

'The continuum hypothesis CH is true in the real universe of sets'

is

$\quad$(42) meaningful $\quad$ (35) meaningless

$\ \ $B. (To be answered only if your answer to IIA is 'meaningful')

$\ \ \ \ \ \ $(2) I think CH is almost certainly true
$\ \ \ \ \ \ $(2) I think CH is more likely true than false
$\ \ \ \ \ \ $(12) I think CH is more likely false than true
$\ \ \ \ \ \ $(14) I think CH is almost certainly false
$\ \ \ \ \ \ $(12) I have no idea whether CH is true or false.
$\ \ $B'. (To be answered only if your answer to IIA is 'meaningless')

$\ \ \ \ $(1) My position on IIA

$\quad\quad$(2) does$\quad$(33) does not

$\ \ \ \ \ \ $cast doubt in my own mind on the value of set theory.

$\ \ \ \ $(2) I am inclined to think that set theory based on the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is destined to play in the long-range future develop-
$\ \ \ \ \ \ \ \ \ $ment of mathematics a

$\ \ \ \ \ \ \ \ \ $(11) more important role than
$\ \ \ \ \ \ \ \ \ $(13) role of equal importance with
$\ \ \ \ \ \ \ \ \ $(11) less important role than

$\ \ \ \ \ \ \ \ \ $set theory based on the denial of the continuum hypothesis.

$\ \ $ C. Assuming that human mathematicians still exist then, I believe that
$\ \ \ \ \ \ \ $in 2067 the prevailing opinion among them will be that the continuum
$\ \ \ \ \ \ \ $problem:

$\ \ \ \ $(4) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the continuum hypothesis is
$\ \ \ \ \ \ \ \ \ $a consequence
$\ \ \ \ $(18) has been settled by the discovery of generally accepted new
$\ \ \ \ \ \ \ \ \ $axioms or methods of proof of which the denial of the continuum
$\ \ \ \ \ \ \ \ \ $hypothesis is a consequence
$\ \ \ \ $(37) has been settled by the general acceptance of the belief that
$\ \ \ \ \ \ \ \ \ $there is no one true set theory and that the continuum hypothesis
$\ \ \ \ \ \ \ \ \ $simply holds in some theories and fails in others
$\ \ \ \ $(11) is still unsettled

III. A. I believe that there is an absolute sense in which every sentence of
$\ \ \ \ \ \ \ \ \ $first-order number theory based on addition, multiplication, and
$\ \ \ \ \ \ \ \ \ $exponentiation is either true or false.

$\quad$(54) yes $\quad$ (26) no

$\ \ \ \ $B. I believe that there is an absolute sense in which every $\underline{\text{universal}}$
$\ \ \ \ \ \ \ \ \ $sentence of first-order number theory based on addition, multiplication,
$\ \ \ \ \ \ \ \ \ $and exponentiation is either true or false.

$\quad$(62) yes $\quad$ (18) no

Please do not sign your ballot.

August 1, 1967
University of California, Los Angeles