I definitely do still intend to read On the Historicity, but I simply have not had the time to give to it, yet.

That said, I’ll definitely note that I’m quite skeptical that Carrier can demonstrate any significant pre-Christian belief in a celestial being named Jesus, let alone one which came to be Euhemerized in the stories of Jesus of Nazareth.

I will certainly agree that there are Messianic traditions surrounding Joshua son of Jehozadak, particularly as relates to Zechariah’s description of him. However, “Messianic” does not imply any sort of divinity, despite Christian misunderstandings of the term. Moreover, there would have been no need to Euhemerize the figure of Joshua son of Jehozadak since he was already known to have been a human being who lived at the beginning of the Second Temple period after coming out of the Babylonian exile. You don’t need to create myths about an earthly life for a being which already had an earthly life.

As for his argument resting on “hundreds” of lines of evidence, even if I grant this rather ludicrous exaggeration, the fact of the matter is that his entire claim is that the gospels are Euhemerizing a pre-Christian divine (or semi-divine) figure from Jewish mythology named Jesus. If Dr. Carrier cannot establish that there WAS a pre-Christian divine (or semi-divine) figure from Jewish mythology named Jesus, then the rest of his lines of evidence are fairly worthless, no matter how many in number. If he cannot demonstrate this foundational part of his claim, the rest of his hypothesis– “hundreds” of lines of evidence and all– is simply ad hoc and specious.

]]>Okay, but I’m pretty sure that’s what he meant. He’s talking about collections that are actually infinite. So, that seems nit-picky. But I follow you know. So, an actual infinite is a number. A transfinite number. A countable number, in some cases, I assume.

“I think that you and I will agree that Even-ness is not a number; however, a number CAN be even and there exist a multitude of even numbers. I could quite easily say that I have two numbers, x and y, such that x<y despite the fact that both x and y are even. None of this sounds unreasonable, right?"

Right.

"In exactly the same way, I might have some numbers, X and Y, such that X<Y despite the fact that both X and Y are infinite."

I mean, I guess it depends. I don't know what "<" means. Cardinality? Size? Quantity?

"As to precisely what is meant by an "infinite number" depends on the particular number system being used. However, in general, a number which is infinite has an absolute value which is greater than any Natural number."

Yea, that's why it's a transfinite number.

"Imagine, for a moment, I told you that I have some number, x, which is positive. Without having to do any calculations and without knowing anything more about the exact value of x, you immediately know that x is greater than every negative number. In exactly the same way, if I have some positive infinite number, X, I do not need to know anything more about the exact value of X to know that it is greater than any Natural number."

Yep.

"I hope this helps. If it's still not completely clear, I'll be happy to address any further questions!"

Okay. I just don't know why his argument is bad. I don't care if the mathematics has been inordinately successful. That's not his point. Craig agrees with this. Mathematically, it's fine. How does this have anything to do with metaphysical possibility? The 'imaginary number' point doesn't do anything for me. Use imaginary numbers all you want. In mathematics, that's great! I have no idea what it means to some 'real-world' counterpart to an imaginary number. But even if there were, I don't see the analogy between imaginary numbers and actual infinites. I can imagine some world where there's an actual infinite collection of stuff; I can't imagine a 'real-world' counterpart for imaginary numbers. The analogy just has to do with the logical/metaphysical possibility distinction. That's all.

"There exist systems of mathematics in which infinite numbers can be subtracted from infinite numbers perfectly consistently– for example, on Surreal numbers (Conway) or Hyperreal numbers (Robinson). These do not lead to the purported contradictions espoused by apologists."

I don't see the relevance of this at all. Great. Surreal and Hyperreal numbers can be subtracted and divided. Awesome. I think everyone agrees you can do this in the world of mathematics. This just ignores the metaphysical-possibility point. I really can't see how you avoid contradictions either way. Maybe I'm dumb. I admit. The only difference between Set Theory and these other numbers is that Set Theory prohibits subtraction and division of infinite sets. The contradictions come from actually subtracting/dividing in the real world. So, if the Hyperreals and the Surreals permit this, all the better for bringing about real-world contradictions/absurdities.

"This bald assertion is an unfortunate bit of question begging. One of the primary definitions of an actually infinite set is a set which contains a proper subset of equal cardinality (Katz 792-795). Belief Map offers no good reason to accept the claim that proper parts always contain less than wholes. One might as well argue that actual infinities can’t exist because actual infinities can’t exist."

Bald assertion? I'm sorry, but this is the intuitive position here. Yes, the part and the whole have equal cardinality. That's not the issue. It's size/quantity. You're being unclear here again. Did he mention this in terms of cardinality? If so, show me where. And further. The 'equal cardinality' thing is what leads to absurd results in the real world. I don't care what happens in the mathematical world. Do all you want there if the rules of mathematics permit it. That's not the argument. It's the metaphysical possibility point that's the argument. Do you know the difference between the two? If you do, I can't tell by what you've written.

"The infinite tug of war is actually just a restatement of the question of subtracting one infinite number from another, which we’ve already discussed. It makes precisely the same mistake as before, treating “infinity” as a number and not recognizing that there are numerous infinite numbers, not all of which are equal. As such, it is therefore easily resolved by proper mathematics."

Ha, ha. I just find this uncharitable. He means an actual infinite! And again, you say 'not equal'. Right! Spell out what you mean. Quantity/size, correct? Resolved by proper mathematics . . . right. That's not the point. You don't seem to understand mathematics possibility and metaphysical possibility.

"The infinite hotel illustrates a counter-intuitive property of actual infinities, but it does not illustrate a metaphysical impossibility or a contradiction. The only way one might legitimately claim that this is an absurdity would be to already reject the possibility of actual infinites. However, since this is being utilized as an argument in support of just such a rejection, to do so would simply be fallaciously circular question begging."

I want you to define 'metaphysical possibility' for me. Because it makes no sense to say I need to reject the possibility of actual infinites because I reject the metaphysical possibility of actual infinites. You're being imprecise again. What do you mean by "reject the possibility"? What do you mean by possibility? Please be more clear. I can reject it in one sense, and affirm it in the mathematical sense. Are you a mathematical formalist? If you are, that needs argument, or you're begging some questions.

"The infinite popsicle does present something of an absurdity. I’ll agree that Bernardete’s scenario is metaphysically impossible, but not for the reason which he suggests. Popsicles are composed of atoms. Atoms have a significantly non-infinitesimal volume. As such, one cannot create a popsicle with an infinite number of layers in 4 cubic inches (or any other finite volume) of space. This thought experiment doesn’t work, but not due to any metaphysical absurdity relating to infinity."

So, here you agree this is metaphysically impossible. I want you to define this please. You appeal to atoms. This isn't strictly metaphysical possibility, though it involves it. This is physical impossibility. It's a physical fact that atoms have the volume they do. So, please define metaphysical possibility for me.

]]>I like to use the concept of Even-ness to be a direct parallel of what I mean when I say that “infinity is a quality of numbers.” A number can be even, but a number cannot be Even-ness. Similarly, a number can be infinite, but a number cannot be infinity.

I think that you and I will agree that Even-ness is not a number; however, a number CAN be even and there exist a multitude of even numbers. I could quite easily say that I have two numbers, x and y, such that x<y despite the fact that both x and y are even. None of this sounds unreasonable, right?

In exactly the same way, I might have some numbers, X and Y, such that X<Y despite the fact that both X and Y are infinite.

As to precisely what is meant by an "infinite number" depends on the particular number system being used. However, in general, a number which is infinite has an absolute value which is greater than any Natural number.

Imagine, for a moment, I told you that I have some number, x, which is positive. Without having to do any calculations and without knowing anything more about the exact value of x, you immediately know that x is greater than every negative number. In exactly the same way, if I have some positive infinite number, X, I do not need to know anything more about the exact value of X to know that it is greater than any Natural number.

I hope this helps. If it's still not completely clear, I'll be happy to address any further questions!

]]>This is really confusing to me. On the one hand, you say that an actual infinite is an IDEA. On the other hand, you’re saying that it’s the idea that a complete set contains a NUMBER of elements greater than any natural number. So, I guess you’re saying it’s an idea, not a number —- even though it’s an idea about a number?

THEN, you’re saying that it’s a QUALITY of numbers, NOT a number itself. But then you say a NUMBER can be INFINITE? Huh? And that there’s a MULTITUDE of infinite NUMBERS. I’m so lost. How on earth can an actual infinite NOT be a number, and yet there can be (1) a NUMBER that is INFINITE, (2) a MULTITUDE of INFINITE NUMBERS, and (3) a NUMBER of elements greater than any natural number????

Please be more precise with your language because it sounds to a laymen like me that you’re just contradicting yourself.

]]>If one rejects the idea of an infinite set, as Dr. Wildberger does, then there does not seem to be any good way to rigorously define what we mean by irrational numbers.

So, using the sqrt(2) example, Wildberger would say that we are only justified in saying that there exists no Rational number equal to sqrt(2); and he argues that we are NOT justified in saying that because of this, there must be some number equal to sqrt(2) which is not Rational.

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