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> If you’re still tempted to quibble, then consider the following parallel question: > Let n equal 3 if God exists, or 5 if God does not exist. Is n prime? Su

> If you’re still tempted to quibble, then consider the following parallel question: > Let n equal 3 if God exists, or 5 if God does not exist. Is n prime? Sure, I'll happily quibble! You're using excluded middle to assert that n is either 3 or 5, but you haven't justified that excluded middle holds for the proposition "God exists".

coldtea 2y ago
>Let n equal 3 if God exists, or 5 if God does not exist. Is n prime? Also whether n is prime is up to the will of God, if God exists. A God is not necessarily bound to the "laws of physics" or even basic logical necessities (a God that does comes from a specific line of theological reasoning, not the general case). He could even make 6 odd if He so wished - altering all math and logical consistency and the whole universe, or just make just 77 even and every other number odd, making so every mathematician finds the new arrangement perfectly consistent and consider it to always have been the valid one! So the answer does kind of "depend on your religious beliefs".
- cwillu 2y ago
  If god creates a system of logic wherein 3 is not prime, he's welcome to, but creating a new system doesn't affect the old one in the slightest.
  - coldtea 2y ago
    >but creating a new system doesn't affect the old one in the slightest. The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those. In fact the current system of odd/even - as well as any other system and law, including the "excluded middle" - can be understood to be God's arbitrary creation. You consider it logically necessary and absolute just because God willed it so, you'd be considering a different one with any arbitrary change as just as logically necessary and absolute if/when God willed it :) People create a lesser conception of God, with specific boundaries and limitations, and then gloat how he is limited. Well, let's instead start with the more common conception of God as limitless and with total arbitrary power.
    - simonh 2y ago
      >The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those. That entirely depends on your specific god belief. Suppose someone believes that god is the eternal principle of consistency? >Well, let's instead start with the more common conception of God as limitless and with total arbitrary power. If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness? You quickly get to then point where the concept ceases having any meaning.
      - coldtea 2y ago
        >That entirely depends on your specific god belief My sentiments exactly. The whole point of the comment was that the part saying "The answer does not depend on your religious beliefs" was wrong. >If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness? Yes. Trivially so. And He can even make it so godlessness involves the presense of God too, without there being any logical inconsistency even (since he's so powerful he shapes logic, not the other way around).
        TheOtherHobbes 2y ago
        So the answer to "P=NP?" is "Let's see what mood god is in today"?
        zbyszek 2y ago
        The answer is "God knows!"
        bee_rider 2y ago
        I think if you believe in one of the omnipotent gods, yea, that is something you believe.
        2y ago
- nkrisc 2y ago
  > So the answer does kind of "depend on your religious beliefs". Only if you choose to misinterpret the point.
  - coldtea 2y ago
    It's not a matter of interpretation. Except if they wanted to say "The answer, iff your religious beliefs are conveniently tame and constrained, so that your God has limited powers and is bound to respecting excluded middle, the laws of physics and other such constraints, doesn't depend on your religious beliefs".
    - badmonkey101 2y ago
      That's just like your opinion man
      - coldtea 2y ago
        Nope. In a discourse constrained by regular logic, it's a logically consistent argument. Their rules, not mine! Any other movie quotes to put forward in lieu of counter-arguments?
        badmonkey101 2y ago
        Invoking a fictitious god with any arbitrary property doesn't make your argument consistent - other then, that's just like your opinion man
        coldtea 2y ago
        >Invoking a fictitious god with any arbitrary property doesn't make your argument consistent Logical consistency only cares about propositions and their relationship, it doesn't care whether an entity involved in one is fictitious or not.
    - hifromwork 2y ago
      Law of excluded middle is a formal logic thing. If you assume law of excluded middle, you can use it for logical deduction. If you don't assume law of excluded middle, you can't. If your axioms are A->B and A then B, and even god can't change that. In contrast laws of physics are a real world thing, and any omnipotent being can meddle with it, basically by definition.
- iainmerrick 2y ago
  This is a very good (and funny) response. If you're allowed to invoke God ironically in the question, you're allowed to take that seriously (meta-ironically?) in the rebuttal!
denotational 2y ago
In classical logic, LEM is valid. If you’re going to quibble over whether LEM is justifiable in this case, then you need to justify why you’re only concerned about LEM; why not drop ex falso too (Kolmogorov had serious issues with this axiom, and initially considered it to be incompatible with a constructive logic) and work in a paraconsistent logic? Also, depending on the precise formulation of this proposition, it doesn’t necessarily need LEM.
- cobbal 2y ago
  Is there a variant of "there exists a program that prints BB(8000)" that doesn't rely on some sort of axiom of choice? Maybe the intuitionists have a more useful definition of computibility if it doesn't posit that such strange machines exist.
  - School-Cotton 2y ago
    > Is there a variant of "there exists a program that prints BB(8000)" that doesn't rely on some sort of axiom of choice? Why does it rely on the axiom of choice? It’s a consequence of the obvious fact that for every integer n, there exists a program that prints n. This doesn’t require choice.
    - cobbal 2y ago
      Maybe choice was stronger than I needed. Thinking about it more, I think the problem is that BB isn't even definable (that I can see) without LEM. There's an "either the machine halts or it doesn't" baked in to the computation of the integer.
      - School-Cotton 2y ago
        A lot of stuff in math doesn't work the same way without the law of the excluded middle, but it's always assumed unless it's stated explicitly that we're working under some other logical system.
- Smaug123 2y ago
calf 2y ago
FYI in the textbook version, they do say to assume the question is unambiguously binary (Sipser 2nd ed. page 162). It is very astute of you to catch that!
- sudoankit 2y ago
  For those having the 3rd edition, Q. 3.22, Page 190, however in the textbook unlike the blog post, 1 is if life exists on Mars, 0 if not.
  - zeroonetwothree 2y ago
    I remember the Mars question from the class I took 20 years ago.

coldtea 2y ago

>Let n equal 3 if God exists, or 5 if God does not exist. Is n prime? Also whether n is prime is up to the will of God, if God exists. A God is not necessarily bound to the "laws of physics" or even basic logical necessities (a God that does comes from a specific line of theological reasoning, not the general case). He could even make 6 odd if He so wished - altering all math and logical consistency and the whole universe, or just make just 77 even and every other number odd, making so every mathematician finds the new arrangement perfectly consistent and consider it to always have been the valid one! So the answer does kind of "depend on your religious beliefs".

cwillu 2y ago
If god creates a system of logic wherein 3 is not prime, he's welcome to, but creating a new system doesn't affect the old one in the slightest.
- coldtea 2y ago
  >but creating a new system doesn't affect the old one in the slightest. The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those. In fact the current system of odd/even - as well as any other system and law, including the "excluded middle" - can be understood to be God's arbitrary creation. You consider it logically necessary and absolute just because God willed it so, you'd be considering a different one with any arbitrary change as just as logically necessary and absolute if/when God willed it :) People create a lesser conception of God, with specific boundaries and limitations, and then gloat how he is limited. Well, let's instead start with the more common conception of God as limitless and with total arbitrary power.
  - simonh 2y ago
    >The idea of "new doesn't affect old" is an idea based on logical and temporal consistency. God is not bound to those. That entirely depends on your specific god belief. Suppose someone believes that god is the eternal principle of consistency? >Well, let's instead start with the more common conception of God as limitless and with total arbitrary power. If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness? You quickly get to then point where the concept ceases having any meaning.
    - coldtea 2y ago
      >That entirely depends on your specific god belief My sentiments exactly. The whole point of the comment was that the part saying "The answer does not depend on your religious beliefs" was wrong. >If god is capable of anything can god decide to make it such that there is no god? Can god decide to self-limit god's own powers? If god intrinsically encompasses all possibilities does that include the possibility of godlessness? Yes. Trivially so. And He can even make it so godlessness involves the presense of God too, without there being any logical inconsistency even (since he's so powerful he shapes logic, not the other way around).
      - TheOtherHobbes 2y ago
        So the answer to "P=NP?" is "Let's see what mood god is in today"?
        zbyszek 2y ago
        The answer is "God knows!"
        bee_rider 2y ago
        I think if you believe in one of the omnipotent gods, yea, that is something you believe.
      - 2y ago
nkrisc 2y ago
> So the answer does kind of "depend on your religious beliefs". Only if you choose to misinterpret the point.
- coldtea 2y ago
  It's not a matter of interpretation. Except if they wanted to say "The answer, iff your religious beliefs are conveniently tame and constrained, so that your God has limited powers and is bound to respecting excluded middle, the laws of physics and other such constraints, doesn't depend on your religious beliefs".
  - badmonkey101 2y ago
    That's just like your opinion man
    - coldtea 2y ago
      Nope. In a discourse constrained by regular logic, it's a logically consistent argument. Their rules, not mine! Any other movie quotes to put forward in lieu of counter-arguments?
      - badmonkey101 2y ago
        Invoking a fictitious god with any arbitrary property doesn't make your argument consistent - other then, that's just like your opinion man
        coldtea 2y ago
        >Invoking a fictitious god with any arbitrary property doesn't make your argument consistent Logical consistency only cares about propositions and their relationship, it doesn't care whether an entity involved in one is fictitious or not.
  - hifromwork 2y ago
    Law of excluded middle is a formal logic thing. If you assume law of excluded middle, you can use it for logical deduction. If you don't assume law of excluded middle, you can't. If your axioms are A->B and A then B, and even god can't change that. In contrast laws of physics are a real world thing, and any omnipotent being can meddle with it, basically by definition.
iainmerrick 2y ago
This is a very good (and funny) response. If you're allowed to invoke God ironically in the question, you're allowed to take that seriously (meta-ironically?) in the rebuttal!

denotational 2y ago

In classical logic, LEM is valid. If you’re going to quibble over whether LEM is justifiable in this case, then you need to justify why you’re only concerned about LEM; why not drop ex falso too (Kolmogorov had serious issues with this axiom, and initially considered it to be incompatible with a constructive logic) and work in a paraconsistent logic? Also, depending on the precise formulation of this proposition, it doesn’t necessarily need LEM.

cobbal 2y ago
Is there a variant of "there exists a program that prints BB(8000)" that doesn't rely on some sort of axiom of choice? Maybe the intuitionists have a more useful definition of computibility if it doesn't posit that such strange machines exist.
- School-Cotton 2y ago
  > Is there a variant of "there exists a program that prints BB(8000)" that doesn't rely on some sort of axiom of choice? Why does it rely on the axiom of choice? It’s a consequence of the obvious fact that for every integer n, there exists a program that prints n. This doesn’t require choice.
  - cobbal 2y ago
    Maybe choice was stronger than I needed. Thinking about it more, I think the problem is that BB isn't even definable (that I can see) without LEM. There's an "either the machine halts or it doesn't" baked in to the computation of the integer.
    - School-Cotton 2y ago
      A lot of stuff in math doesn't work the same way without the law of the excluded middle, but it's always assumed unless it's stated explicitly that we're working under some other logical system.
Smaug123 2y ago

calf 2y ago

FYI in the textbook version, they do say to assume the question is unambiguously binary (Sipser 2nd ed. page 162). It is very astute of you to catch that!

sudoankit 2y ago
For those having the 3rd edition, Q. 3.22, Page 190, however in the textbook unlike the blog post, 1 is if life exists on Mars, 0 if not.
- zeroonetwothree 2y ago
  I remember the Mars question from the class I took 20 years ago.