Proofs... So what?

[Nirvana-Eld]

After some thinking I am willing to accept that there must have been a first cause that initiated the series of events we call existence. I think that this is the underlying substratum of causuality. Here the problem. So what? I do not see the connection from "there must have been a first cause" to "this is a being which we all call God" (quod omnes dicunt Deum). I see the point of the First Cause Idea more as pointing to a natural process, not a Christian God and or the classic "Omnipotent, Omniscient, Omnibenevolent God". There seems to be no connection.

I would like the proponents of the First Cause Argument who believe in the God of the three "O's" to logically make the connection between a process and a God. Civility is a must! I just glanced through a thread by a "killingevolution" ( ) and I would like this thread to be cleaner. 8)

[harvey1]

But... I've never claimed that the infinite set of regressive causes must be computable in finite time. I'm afraid I still don't see why this is a requirement...

If you use an analogy, that analogy must be as accurate as possible. In the case of a number line, this is not an accurate representation for causes since we are not saying that the numbers are physical occurrences, we are saying they can represent physical occurrences (events in the causal past). However, in order to represent a real physical cause, part of the condition in doing so is that we must be able to match the event and number in a one-to-one comparison as we go back into the causal past (e.g., -1="the window broke," -2="the ball struck the window," -3="the ball approached the window at high velocity," -4="the ball took off into the air," -5="the bat struck the ball right on the money," -6="she swung the bat at the ball," etc.). This is to say that the numbers are computable. If the numbers aren't computable, then you couldn't reference an incomputable number with an event since you couldn't ever arrive at that number to make the one-to-one matching of the event.

What do you mean by "reference an event"? I claim that it's possible to reach any specific event from any other specific event in finite time.

Again, you are computing an infinite collection of causes backward in finite time, and it is not possible to compute an infinite collection. That's why the Axiom of Infinity is required.

For example, we could look at any event happening in the present, and trace a causal chain from it to the "past" and "future" events (causality-wise, not necessarily time-wise), of arbitrary length, as long as that length is finite.

The length of an infinite set is not finite, it's infinite by definition.

[Bugmaster]

If you use an analogy, that analogy must be as accurate as possible. In the case of a number line, this is not an accurate representation for causes since we are not saying that the numbers are physical occurrences, we are saying they can represent physical occurrences (events in the causal past).

Um, I don't think the numbers are physical anythings. When I'm counting apples, I'm mapping each apple to an integer. So, when I say, "I have 5 apples", I'm using the integer 5 to represent the total number of apples (and the biggest apple, if I sort them by size). Why is this a problem ?

However, in order to represent a real physical cause, part of the condition in doing so is that we must be able to match the event and number in a one-to-one comparison as we go back into the causal past (e.g., -1="the window broke,"... This is to say that the numbers are computable.

I disagree. I claim that, given a set of apples with the cardinality of aleph-0, and the set of integer with the cardinality of aleph-0, we can map each apple to an integer, despite the fact that there are infinite apples. I can probably prove this mathematically, if you disagree.

Again, you are computing an infinite collection of causes backward in finite time...

Wrong. I am computing a finite distance between two events. As I've shown previously, if you take any two specific integers (such as 17 and 92), the difference between them is finite, and can be enumerated in finite time. If events can be mapped to integers, it follows that the "causal distance" between any two specific events can be computed in finite time.

I fully agree that it's impossible to exhaustively enumerate all causes of any particular event, because there's an infinite number of them; nor is it possible to enumerate all integers smaller than 7. But this is not the same as saying that we cannot enumerate all integers between two given integers.

You seem to imply that we should be able to exhaustively enumerate all causes for every event in order for this event to be non-random... Why do you think that ? How do you define "random", anyway ?

For example, we could look at any event happening in the present, and trace a causal chain from it to the "past" and "future" events (causality-wise, not necessarily time-wise), of arbitrary length, as long as that length is finite.

The length of an infinite set is not finite, it's infinite by definition.

I said "of arbitrary length". Arbitrary, yet still finite.

[harvey1]

Um, I don't think the numbers are physical anythings... Why is this a problem?

That's what I just said.

I disagree. I claim that, given a set of apples with the cardinality of aleph-0, and the set of integer with the cardinality of aleph-0, we can map each apple to an integer, despite the fact that there are infinite apples. I can probably prove this mathematically, if you disagree.

That's if you treat the numbers as a label that exists as a result of the Axiom of Infinity. If we are to compute the numbers using a finite Turing machine, then you cannot show that there are an infinite number of apples without the Axiom of Infinity.

Again, you are computing an infinite collection of causes backward in finite time...

Wrong. I am computing a finite distance between two events.

If there is a finite distance between any two events in a infinite collection of events, then in what way is it an infinite collection?? Infinite collection of negative integers means that there is an infinite number of members between -1 and those uncomputable members within that set. Infinite means not finite, right?

As I've shown previously, if you take any two specific integers (such as 17 and 92), the difference between them is finite, and can be enumerated in finite time.

That's for computing finite sets. If you are taking any two integers and enumerating the difference you are doing a computation.

I fully agree that it's impossible to exhaustively enumerate all causes of any particular event, because there's an infinite number of them; nor is it possible to enumerate all integers smaller than 7. But this is not the same as saying that we cannot enumerate all integers between two given integers.

That's seems like a contradiction to me. If you have an infinite collection, we agree that you cannot compute all the integers, so we agree that there are uncomputable numbers in an infinite collection, but yet you want to say that there cannot be uncomputable numbers in an infinite collection. How can you not compute every number in a complete infinite set while being able to compute any number in a complete infinite set? That's contradictory.

I said "of arbitrary length". Arbitrary, yet still finite.

It seems to me that your argument hinges upon there being no uncomputable numbers in an infinite collection despite the fact that you acknowledge that you cannot compute every number in an infinite set. If a number is uncomputable then how can you compute the distance from that number to -1?

[Bugmaster]

That's if you treat the numbers as a label that exists as a result of the Axiom of Infinity.

I find this statement a bit weird. How can I treat the set of integers as anything other than a set of integers ? Its definition leaves no room for interpretation.

If we are to compute the numbers using a finite Turing machine, then you cannot show that there are an infinite number of apples without the Axiom of Infinity.

This is false; a more correct statement would be, "If we are to compute the numbers using a finite Turing machine in finite time, then you cannot show that there are an infinite number of apples". This statement is more correct, but it still sounds wrong to me, and I'm not sure how the Axiom of Infinity comes into play here at all. If we define the set of integers as "the set of positive natural numbers U negative natural numbers U {0}", then we can easily arrive at its cardinality being aleph-0 without using any extraneous axioms, other than the basics of set theory.

If there is a finite distance between any two events in a infinite collection of events, then in what way is it an infinite collection?? Infinite collection of negative integers means that there is an infinite number of members between -1 and those uncomputable members within that set. Infinite means not finite, right?
...
That's seems like a contradiction to me. If you have an infinite collection, we agree that you cannot compute all the integers, so we agree that there are uncomputable numbers in an infinite collection

No, this is false. In the set of integers, there are no uncomputable integers (assuming you start from some arbitrary point, such as 0), but there's an infinite number of computable integers.

Think about it this way: are all integers in the set of integers the same (except for their magnitude and sign) ?

If you think that the set of integers contains some uncomputable integers, then, by all means, show them to me. Show me any two integers the difference between which cannot be enumerated, and I'll concede my point. Note that aleph-0 (or any other kind of aleph-N) is not a member of the set of integers.

If a number is uncomputable then how can you compute the distance from that number to -1?

I cannot, but every integer is "computable" by your definition. For each z in Z, there's a finite number of increments (or decrements) that will take us from 0 to z. This number is equal to abs(z).

[harvey1]

That's if you treat the numbers as a label that exists as a result of the Axiom of Infinity.

I find this statement a bit weird. How can I treat the set of integers as anything other than a set of integers? Its definition leaves no room for interpretation.

That is, as a set the integers are not treated as having been computed but as having been shown how to compute them in principle.

If we are to compute the numbers using a finite Turing machine, then you cannot show that there are an infinite number of apples without the Axiom of Infinity.

This is false; a more correct statement would be, "If we are to compute the numbers using a finite Turing machine in finite time, then you cannot show that there are an infinite number of apples".

BM, if a finite Turing machine will not halt the computation is "uncomputable." (This use of the term uncomputable should not to be confused with how this term is traditionally designated, so I should say "not computable.") If a Turing machine were to count the number of an infinite number of apples, the operation would not halt. Do you agree?

If we define the set of integers as "the set of positive natural numbers U negative natural numbers U {0}", then we can easily arrive at its cardinality being aleph-0 without using any extraneous axioms, other than the basics of set theory.

The Axiom of Infinity is an axiom of ZF set theory.

but there's an infinite number of computable integers.

If we let a Turing machine run for aleph-zero years, then all integers are computable integers. But, in order to say that there is a set of infinite number of years you need the Axiom of Infinity.

If you think that the set of integers contains some uncomputable integers, then, by all means, show them to me. Show me any two integers the difference between which cannot be enumerated, and I'll concede my point. Note that aleph-0 (or any other kind of aleph-N) is not a member of the set of integers.

I can't show you because they need an infinite number of digits to type, and that would take an infinite amount of years to write them (and you would need an infinite amount of years to read the number)!