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to bring together some of the most prominent thinkers who have struggled with the following questions:

 

(1) Do the questions that are independent of the standard axioms admit of determinate answers?

 

(2) If so then what are those answers and how might we go about determining them?

Mathematics as a professional activity with serious numbers of workers is quite new, let’s say, one hundred years old, although even that is a stretch. Assuming that the human race thrives, what is this compared to, say, a thousand more years? It is probably merely a bunch of simple observations in comparison.

 

Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Gödel phenomena.

 

We can, of course, take this even further. A million years’ time is absolutely nothing in astronomical time. Our sun has several billion good years left (although the sun will cause a lot of global warming). Mathematics in a billion years’ time? Who can know what that will be like. However, I am convinced that the Gödel legacy will remain very much alive – at least as long as there is vibrant mathematical activity.

to bring together some of the most prominent thinkers who have struggled with the following questions:

(1) Do the questions that are independent of the standard axioms admit of determinate answers?

(2) If so then what are those answers and how might we go about determining them?

Mathematics as a professional activity with serious numbers of workers is quite new, let’s say, one hundred years old, although even that is a stretch. Assuming that the human race thrives, what is this compared to, say, a thousand more years? It is probably merely a bunch of simple observations in comparison.

Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Gödel phenomena.

We can, of course, take this even further. A million years’ time is absolutely nothing in astronomical time. Our sun has several billion good years left (although the sun will cause a lot of global warming). Mathematics in a billion years’ time? Who can know what that will be like. However, I am convinced that the Gödel legacy will remain very much alive – at least as long as there is vibrant mathematical activity.

Feferman is ultimately skeptical of the proram, he believes that there is no hard, pure mathematical reason for new axioms and that therefore the question is only philosophical in nature and thus has no definitive answer. More on Feferman's position (especially on his attitude that the continuum hypothesis is not a definite mathematical problem and is inherently vague) will be referenced further down.

Steel is of the position that there still major open questions in set theory that have no been resolved and therefore Gödel's program cannot be shown to be defeated. He argues that there are still outstanding problems in descriptive set theory, as well as problems like the continuum hypothesis, that we have proven independent of ZFC and therefore, of course, we need new axioms to settle them.

Again, Joel Hamkin has provided an answer which outlines his stance in the pluralism vs. non pluralism debate and I agree with his suggestion to follow the links he provided. Here is another link to his The set-theoretic multiverse paper, and Koellner's response from the project Hamkins on the Multiverse.

Finally the last of my highlights is once again John Steel's work. His four slides The Triple Helix, Gödel's program, Gödel's Legacy, and Generic Absoluteness and the Continuum Problem (I won't link them because they are automatic downloads instead of being hosted on Harvard's server, but they are available on the webpage under his name) give great historical overview of Gödel's large cardinal program, it's prominence in contemporary set theory and it's lasting philosophical impact.

Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Go ̈del phenomena.

Feferman is ultimately skeptical of the program, he believes that there is no hard, pure mathematical reason for new axioms and that therefore the question is only philosophical in nature and thus has no definitive answer. More on Feferman's position (especially on his attitude that the continuum hypothesis is not a definite mathematical problem and is inherently vague) will be referenced further down.

Steel is of the position that there are still major open questions in set theory that have not been resolved and therefore Gödel's program cannot be shown to be defeated. He argues that there are still outstanding problems in descriptive set theory, as well as problems like the continuum hypothesis, that we have proven independent of ZFC and therefore, of course, we need new axioms to settle them.

Again, Joel Hamkins has provided an answer which outlines his stance in the pluralism vs. non pluralism debate and I agree with his suggestion to follow the links he provided. Here is another link to his The set-theoretic multiverse paper, and Koellner's response from the project Hamkins on the Multiverse.

Finally the last of my highlights is once again John Steel's work. His four slides The Triple Helix, Gödel's program, Gödel's Legacy, and Generic Absoluteness and the Continuum Problem (I won't link them because they are automatic downloads instead of being hosted on Harvard's server, but they are available on the webpage under his name) give great historical overview of Gödel's large cardinal program, its prominence in contemporary set theory and its lasting philosophical impact.

Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Gödel phenomena.

This topic has been debated and written about by many mathematical logicians, philosophers of mathematics, as well as contemporary set theorists who do not consider their work to be strictly within the field of logic any longer. An important piece of the debate is contained within the article Does Mathematics Need New Axioms?, which is a collection of four essays and rebuttals on this topic written by Solomon Feferman, Penelope Maddy, John Steel, and Harvey Friedman and presented at the Annual ASL meeting in 2000. From the abstract:

This topic has been debated and written about by many mathematical logicians, philosophers of mathematics, as well as contemporary set theorists who do not consider their work to be strictly within the field of logic any longer. An important piece of the debate is contained within the article Does Mathematics Need New Axioms?, which is a collection of four essays and rebuttals on this topic written by Solomon Feferman, Penelope Maddy, John Steel, and Harvey Friedman and presented at the Annual ASL meeting in 2000. From the abstract: