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edited Mar 25 2012 at 19:02

I would like to add to the thoughtful answers to this question and also respond to a couple points made in the answers. Please note that this is mostly a philosophical answer.

The part of the question that I would like to address is

. . . it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory

It might seem like since there are a variety of universes, that some of them must be flawed. In particular that since there are universes which are models of $CH$ and $\neg CH$ that some of these universes must be flawed. The accepted answer to this question has shown that it is coherent to resolve this difficulty in thinking by calling conflicting universes or sets illusions. However, one can overcome this apparent cognitive difficulty by living locally in the multiverse. For the purpose of understanding how to see through the proper perspective in the set-theoretic multiverse, let me create some characters, if you will tolerate, which live in the multiverse. Suppose in the multiverse there are travelers and inhabitants. Travelers like to travel far and wide and often, while inhabitants like to stay home. Let's go to a universe $V$ which models $GCH$. Inhabitants in this universe know that the continuum hypothesis is true. However, they also know that there are many travelers coming to visit their universe. An inhabitant meets a traveler who has just arrived from another universe $W$ where the continuum is $\aleph_{10}$. The traveler sees that in universe $V$ the continuum is $\aleph_1$, and so accepts that the continuum is now a different size because she is in a new place. The inhabitant is having a more difficult time understanding since he has not traveled beyond his native universe. The traveler tells him that in universe $W$ the size of the real numbers is $\aleph_{10}$, but he doesn't understand because the size of the real numbers is most definitely $\aleph_1$, and he has no "experience of the contrary". So, the traveler decides to take him with her to another universe $X$ where the continuum is $\aleph_2$. Via a generic filter and forcing relation, they travel to universe $X$ where the inhabitant of universe $V$ can now see with his own eyes that in this world the continuum is indeed the second uncountable cardinal! So, truth in the multiverse depends on location. This way of understanding perspective is discussed in detail in Hamkins' paper on the set-theoretic multiverse, and in the paper regarding a dream solution to CH which is quoted below:

Part of my goal in the multiverse article was to tease apart two often-blurred aspects of set-theoretic Platonism, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.

Second, I would like to respond to Andrej's perspective that all universes are created equal. I think that there are places in the multiverse which are more pleasant than others depending on one's preferences. For example, the inhabitants of a universe where $GCH$ holds might believe that their universe is the best since there are so many travelers who have a lay-over there. I like this type of universe very much and indeed in my experience with forcing, it is very helpful (in order to count in the ground model) to have the $GCH$ hold. I have also been to a universe where Martin's Axiom holds and I really liked doing mathematics in that universe. However, some regions of the multiverse may be less appealing, say a universe without the axiom of choice.

Finally, I would like to address the second to last paragraph of Joel's response. He says that it is crippling to have to consider the rich set-theoretic multiverse as a mere simulation, an illusion we experience in the universe. I agree, but I think it is important to discuss and distiguish the difference between the dreams of the universe and the reality of the multiverse. In every universe, there are the classes of names of other universes with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent. It is only in the presensce of a generic filter that this dream class can become a real universe. What I am saying is that even if someone holds the view that the multiverse is really an illusion experienced in the universe then that person could not be talking about the actual multiverse, but only a reflection.

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edited Mar 22 2012 at 22:33

Erin K Carmody
1,143●4●15

I would like to add to the thoughtful answers to this question and also respond to a couple points made in the answers. It should be noted Please note that this is mostly a philosophical answer.

The part of the question that I would like to address is

. . . it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory

Part of my goal in the multiverse article was to tease apart two often-blurred aspects of set-theoretic Platonism, namely, to separate the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.

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edited Mar 22 2012 at 21:37

Erin K Carmody
1,143●4●15

I would like to add to the thoughtful answers to this question and also respond to a couple points made in the answers. It should be noted that this is mostly a philosophical answer.

The part of the question that I would like to address is

. . . it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory

It might seem like since there are a variety of universes, that some of them must be flawed. In particular that since there are universes which are models of $CH$ and $\neg CH$ that some of these universes must be flawed. The accepted answer to this question has shown that it is coherent to resolve this difficulty in thinking by calling conflicting universes or sets illusions. However, one can overcome this apparent cognitive difficulty by living locally in the multiverse. For the purpose of understanding how to see through the proper perspective in the set-theoretic multiverse, let me create some characters, if you will tolerate, which live in the multiverse. Suppose in the multiverse there are travelers and inhabitants. Travelers like to travel far and wide and often, while inhabitants like to stay home. Let's go to a universe $V$ which models $GCH$. Inhabitants in this universe know that the continuum hypothesis is true. However, they also know that there are many travelers coming to visit their universe. An inhabitant meets a traveler who has just arrived from another universe $W$ where the continuum is $\aleph_{10}$. The traveler sees that in universe $V$ the continuum is $\aleph_1$, and so accepts that the contiuum continuum is now a different size because she is in a new place. The inhabitant is having a more difficult time understanding since he has not traveled beyond his native universe. The traveler tells him that in universe $W$ the size of the real numbers is $\aleph_{10}$, but he doesn't understand because the size of the real numbers is most definitely $\aleph_1$, and he has no "experience to of the contrarycontrary". So, the traveler decides to take him with her to another universe $X$ where the continuum is $\aleph_2$. Via a generic filter and forcing relation, they travel to universe $X$ where the inhabitant of universe $V$ can now see with his own eyes that in this world the continuum is indeed the second uncountable cardinal! So, truth in the multiverse depends on location. This way of understanding perspective is discussed in detail in Hamkins' paper on the set-theoretic multiverse, and in the paper regarding a dream solution to CH which is quoted below:

Part of my goal in the multiverse article was to tease apart two often-blurred aspects of set-theoretic Platonism, namely, to separate teh the claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.

Finally, I would like to address the second to last paragraph of Joel's response. He says that it is crippling to have to consider the rich set-theoretic multiverse as a mere simulation, an illusion we experience in the universe. I agree, but I think it is important to discuss and distiguish the difference between the dreams of the universe and the reality of the multiverse. In every universe, there are the classes of names of other universes with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent. It is only in the presense presensce of a generic filter that this dream class can become a real universe. What I am saying is that even if someone holds the view that the multiverse is really an illusion experienced in the universe then that person could not be talking about the actual multiverse, but only a reflection.

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answered Mar 22 2012 at 20:13

Erin K Carmody
1,143●4●15

I would like to add to the thoughtful answers to this question and also respond to a couple points made in the answers. It should be noted that this is mostly a philosophical answer.

The part of the question that I would like to address is

. . . it would seem that from this perspective that the CH worlds are already flawed and that to defend CH against not-CH one would have to say that the existence of 'Cohen reals' in the not-CH worlds is somehow illusory

It might seem like since there are a variety of universes, that some of them must be flawed. In particular that since there are universes which are models of $CH$ and $\neg CH$ that some of these universes must be flawed. The accepted answer to this question has shown that it is coherent to resolve this difficulty in thinking by calling conflicting universes or sets illusions. However, one can overcome this apparent cognitive difficulty by living locally in the multiverse. For the purpose of understanding how to see through the proper perspective in the set-theoretic multiverse, let me create some characters, if you will tolerate, which live in the multiverse. Suppose in the multiverse there are travelers and inhabitants. Travelers like to travel far and wide and often, while inhabitants like to stay home. Let's go to a universe $V$ which models $GCH$. Inhabitants in this universe know that the continuum hypothesis is true. However, they also know that there are many travelers coming to visit their universe. An inhabitant meets a traveler who has just arrived from another universe $W$ where the continuum is $\aleph_{10}$. The traveler sees that in universe $V$ the continuum is $\aleph_1$, and so accepts that the contiuum is now a different size because she is in a new place. The inhabitant is having a more difficult time understanding since he has not traveled beyond his native universe. The traveler tells him that in universe $W$ the size of the real numbers is $\aleph_{10}$, but he doesn't understand because the size of the real numbers is most definitely $\aleph_1$, and he has no experience to the contrary. So, the traveler decides to take him with her to another universe $X$ where the continuum is $\aleph_2$. Via a generic filter and forcing relation, they travel to universe $X$ where the inhabitant of universe $V$ can now see with his own eyes that in this world the continuum is indeed the second uncountable cardinal! So, truth in the multiverse depends on location. This way of understanding perspective is discussed in detail in Hamkins' paper on the set-theoretic multiverse, and in the paper regarding a dream solution to CH which is quoted below:

Part of my goal in the multiverse article was to tease apart two often-blurred aspects of set-theoretic Platonism, namely, to separate teh claim that the set-theoretic universe has a real mathematical existence from the claim that it is unique.

Finally, I would like to address the second to last paragraph of Joel's response. He says that it is crippling to have to consider the rich set-theoretic multiverse as a mere simulation, an illusion we experience in the universe. I agree, but I think it is important to discuss and distiguish the difference between the dreams of the universe and the reality of the multiverse. In every universe, there are the classes of names of other universes with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent. It is only in the presense of a generic filter that this dream class can become a real universe. What I am saying is that even if someone holds the view that the multiverse is really an illusion experienced in the universe then that person could not be talking about the actual multiverse, but only a reflection.