Not necessarily. Even though PI isn't a rational number, it doesn't mean that it necessarily contains any given subsequence. Let me take another non-rational number:
0.12112111211112...
It won't ever contain the prime number 3, nor will it contain the complete works of Shakespeare. PI is similarly constrained to a specific pattern. I agree that an infinite un-patterned sequence will contain such sequences, but whether you include or exclude the axiom of choice will determine whether such numbers exist or not.
Whether or not PI actually contains all primes as subsequences, I don't know. I'd suggest you present a proof one way or the other.
I'm quite happy that Apple chose something other than Gecko, even though I love using the current Mozilla browser, and think that the current core is fairly good. The reason is that it means that web developers will hopefully start to notice that there are lots more users who ARENT IE or Netscape 4.x, and thus start using the actual standards, rather than the abominations that IE and Netscape 4.x implement.
>People who are certain are a large part of the problem. WHENEVER you are certain, you've made a mistake.
Then, I guess you are part of the problem yourself, because you stated that very unambiguously, and as a certainty, not as a possibility, and thus, you yourself have made the same mistake that you are pointing out.
The problem is that without some base domain that one considers true (the set of axioms one believes), or that one can be certain about, you can't even talk about things being true or false, or even probable or improbable.
A better statement of this type of statement is that many people make their set of axioms their only category for true/false, rather than having a set of axioms that they believe are unconditionally true, and then another set of things they believe are definitionally true, another that they believe are probabilistically true etc. Another common problem is that they do have those sets, but they tend to put too many things into the set of things they believe axiomatically, rather than having a tiny (minimal) set of axioms, and then having everything deriveable from those in other sets of belief.
This type of thinking will be very familiar to anyone who has derived algebra theorems etc from a base set of axioms that are made as small as possible, except that in real life, one needs more than just true, false and hypothesis, one needs to actually grade the hypotheses based on experience, because the set of known to be true items isn't large enough to actually live life with.
That is where faith comes in, that the universe is a reasonable place, and is understandable, and such faith is as much needed for science as it is for believing in God or gods. Ultimately, whatever you decide to put into the axiomatic set is based on faith (actually, even that there can be such a set ends up being based on faith, or "belief".) That isn't to say that people don't change what they put into the axiomatic set based on their experiences, but you still end up believing that you actually had those experiences etc, in order for the experience to modify the belief. There are religions and people who disagree with me on this, that the universe doesn't make sense or is an illusion, and there just isn't any way to argue about it, because we don't even agree on the basic concepts.
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Not necessarily. Even though PI isn't a rational number, it doesn't mean that it necessarily contains any given subsequence. Let me take another non-rational number: 0.12112111211112... It won't ever contain the prime number 3, nor will it contain the complete works of Shakespeare. PI is similarly constrained to a specific pattern. I agree that an infinite un-patterned sequence will contain such sequences, but whether you include or exclude the axiom of choice will determine whether such numbers exist or not. Whether or not PI actually contains all primes as subsequences, I don't know. I'd suggest you present a proof one way or the other.
I'm quite happy that Apple chose something other than Gecko, even though I love using the current Mozilla browser, and think that the current core is fairly good. The reason is that it means that web developers will hopefully start to notice that there are lots more users who ARENT IE or Netscape 4.x, and thus start using the actual standards, rather than the abominations that IE and Netscape 4.x implement.
Then, I guess you are part of the problem yourself, because you stated that very unambiguously, and as a certainty, not as a possibility, and thus, you yourself have made the same mistake that you are pointing out.
The problem is that without some base domain that one considers true (the set of axioms one believes), or that one can be certain about, you can't even talk about things being true or false, or even probable or improbable.
A better statement of this type of statement is that many people make their set of axioms their only category for true/false, rather than having a set of axioms that they believe are unconditionally true, and then another set of things they believe are definitionally true, another that they believe are probabilistically true etc. Another common problem is that they do have those sets, but they tend to put too many things into the set of things they believe axiomatically, rather than having a tiny (minimal) set of axioms, and then having everything deriveable from those in other sets of belief.
This type of thinking will be very familiar to anyone who has derived algebra theorems etc from a base set of axioms that are made as small as possible, except that in real life, one needs more than just true, false and hypothesis, one needs to actually grade the hypotheses based on experience, because the set of known to be true items isn't large enough to actually live life with.
That is where faith comes in, that the universe is a reasonable place, and is understandable, and such faith is as much needed for science as it is for believing in God or gods. Ultimately, whatever you decide to put into the axiomatic set is based on faith (actually, even that there can be such a set ends up being based on faith, or "belief".) That isn't to say that people don't change what they put into the axiomatic set based on their experiences, but you still end up believing that you actually had those experiences etc, in order for the experience to modify the belief. There are religions and people who disagree with me on this, that the universe doesn't make sense or is an illusion, and there just isn't any way to argue about it, because we don't even agree on the basic concepts.