(February 26, 2018 at 8:37 am)polymath257 Wrote:

(February 26, 2018 at 4:33 am)Grandizer Wrote: Not adding any layer on top of what is clearly an implication of the B-theory of time. Maybe it's the word "illusion" that puts you off, but if you want "causality" to still be implied by the B-theory of time, you do have to redefine the word to mean something that doesn't indicate the flow of time or actual change or actual motion. Fresh fruit does not cause rotten fruit in the sense that it morphs into it; the "antecedent" fresh fruit is still there and the "consequent" rotten fruit is there as well (just in different time moments). Baby horses do not actually eventually morph into adult horses; the baby horse still exists along with the adult version of it. And the key reason we have this one perceived direction happening in this one particular local universe is because of the increase of entropy from one state of this universe to the next, with entropy being at its lowest in the moment closest to the supposed "Big Bang singularity". But otherwise, as far as the laws of physics themselves are concerned, there is no direction of time. And even then, direction does not mean temporal causation anyway; both "cause" and "effect" simultaneously and eternally exist.



No, it isn't. The elements have always been in such a set. How many times do you need to be told this?

I'm going to push back very slightly here. Even in general relativity, which regards spacetime as a single manifold and time does not flow (so B theory is closest), there is a notion of causality.  There are constraints on how things can be at different time slices. So, we will find the rotten apple time slice after that of the fresh apple and not before (in terms of the time coordinate). We find higher entropy states at later time slices than lower entropy states, etc.

Time is still all 'out there'. It does not 'flow' in the sense of the A theory. But there are still patterns between the time slices and we call *those* patterns causality.

Sure, but as you just said/implied, direction of entropy is what's providing an "objective" direction to time here. If entropy was somehow equal (hypothetically speaking) between all the timeslices, then we wouldn't be able to tell which slice causes which slice.

Quote:https://en.wikipedia.org/wiki/Graham%27s_number

I am curious if Steve thinks it is possible for Graham's number of objects to exist in our universe.

Oh, not that. Please don't remind me of that number ...

---

Since really what this whole fiasco is about has to do with the Kalam Cosmological Argument itself requiring an infinite past to be impossible, let me remind Steve what William Lane Craig (the main proponent of the argument) has to say about this argument:

Quote:From start to finish, the kalam cosmological argument is predicated upon the A-Theory of time. On a B-Theory of time the universe does not in fact come into being or become actual at the Big Bang; it just exists tenselessly as a four-dimensional space-time block which is finitely extended in the earlier than direction. If time is tenseless, then the universe never really comes into being, and therefore the quest for a cause of its coming into being is misconceived. Although G. W. F. Leibniz's question, Why is there (tenselessly) something rather than nothing? should still rightly be asked, there would be no reason to look for a cause of the universe's beginning to exist, since on tenseless theories of time the universe did not begin to exist in virtue of its having a first event anymore than a meter stick begins to exist in virtue of having a first centimeter. . . . Thus, the real issue separating the proponent of the kalam cosmological argument and critics of the first premiss is the objectivity of tense and temporal becoming.

https://www.reasonablefaith.org/question...-principle

Meaning that WLC himself acknowledges that the KCA relies on the A-theory of time to be true. Otherwise, note the bolded. Surely Steve isn't as great of a philosopher as WLC, so perhaps he should pay attention to what WLC is saying, before making rash assertions here about the B-theory of time.

(February 25, 2018 at 5:32 pm)polymath257 Wrote:

(February 25, 2018 at 2:17 pm)SteveII Wrote: Third, the idea that all events that will ever happen are equally real simultaneously is NOT the same as infinite set theory in mathematics. The first is an actual infinite and the second is a potential infinite. To be clear, mathematicians are talking about potential infinities when the talk about sets. This is because one side is bounded and only the open side is potentially infinite. These two terms have very different definitions and cannot be used interchangeably or use one to prove the other.

This is just false. Mathematicians are talking about actual infinities. They even talk about different sizes of actual infinities (for example the difference between countably and uncountably infinite sets).

There is no requirement of having an 'open end' in the description of a set:

N={x: x is a natural number}

is a perfectly well defined, infinite set. No 'open side' and no 'bounded side'. The description in terms of a list,

N={1,2,3,4,...}

is more a convenience for those who cannot read mathematics than anything else.

What you seem to think is that the second is some sort of process. It isn't. It *does* appeal to your understanding to know what things are in the set and what are not. But the list itself is just a notational convention and nothing else. In the same way, we can define an uncountbaly infinite set

R={x: x is a real number}

or

[0,1]={x: x is a real number and 0<=x<=1}.

Both of these are uncountably infinite sets.

There is no such thing as a 'potentially infinite set' in math. Sets are either finite or infinite. In the latter case, they are actually infinite. They can be countably infinite or uncountably infinite. For the uncountbaly infinite sets, there are infinitely many different possible cardinalities (although very few are used in practice)

Look in *any* math book and you will find NO distinction made between potential and actual infinity. The reason? Those notions have been replaced. They are no longer used because they are confused and ill defined.

So, I will make a challenge. Look in *any* advanced level math book produced in the last half century. Find *any* reference that discusses *at all* the notion of 'potential infinity'. I challenge you to find a single source *in math* for your claims from the last 50 years (I'll even go 75 years).

Okay, I was wrong about what mathematicians consider their sets to be. They do consider them to be actual infinite. But what does that mean? It does not mean the same thing a physicist (or even a philosopher) would mean with the same phrase. 

However, my point was better said in my post from page 6:

Quote:In the philosophy of mathematics, the abstraction [defined below] of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity {emphasis added}

Quote:In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

In fact, regarding the Axiom of Infinity article, later on the article says: 

Quote:Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation...

I am not claiming the merits of one set of axioms or another--only illustrating that axioms are assumptions and are not self-evident truths (more below on this). 

Quote:Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics) {emphasis added}

So, putting braces {} around 1, 2, 3, 4, 5... makes a new mathematical object: A set. This is not a thing found in the real world that they needed to describe. This is an abstract object  created by axiom. More specifically, a non-logical axiom: 

Quote:As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question...in the philosophy of mathematics.[5] https://en.wikipedia.org/wiki/Axiom#Mathematical_logic {emphasis added}

Further, when mathematicians say "exist", they do not mean the same thing as nonmathematicians:

Quote:...The use of the word "exist" is merely a grammatical convenience here; mathematicians and nonmathematicians do not mean quite the same thing by this word. Unfortunately, we mathematicians don't have a better word; to be more precise we would have to replace this one word with entire paragraphs. If we assume the Axiom of Choice, we are not really stating that we believe in the physical "existence" of those sets or functions. Rather, we are stating that (at least for the moment) we will agree to the convention that we are permitted write proofs in a style as though those sets or functions exist.

Whether those sets or functions "really" exist is actually not important, so long as they do not give rise to contradictions. Mathematicians are perfectly willing to use devices that may be fictional, as intermediate steps in getting from a real problem to a real solution. https://math.vanderbilt.edu/schectex/cou...ntial.html

This last one is actually an easy to read article that explains things in plain language. 

THEREFORE, because infinite sets are a results of non-logical axioms (assumptions) that are not self-evidently true AND, the term 'exist' is not the same in mathematics, I stand by my claim that infinite sets in mathematics are not an indication that an actual infinite series or objects can exists in reality.

(February 26, 2018 at 2:43 pm)SteveII Wrote:

(February 25, 2018 at 5:32 pm)polymath257 Wrote: This is just false. Mathematicians are talking about actual infinities. They even talk about different sizes of actual infinities (for example the difference between countably and uncountably infinite sets).

There is no requirement of having an 'open end' in the description of a set:

N={x: x is a natural number}

is a perfectly well defined, infinite set. No 'open side' and no 'bounded side'. The description in terms of a list,

N={1,2,3,4,...}

is more a convenience for those who cannot read mathematics than anything else.

What you seem to think is that the second is some sort of process. It isn't. It *does* appeal to your understanding to know what things are in the set and what are not. But the list itself is just a notational convention and nothing else. In the same way, we can define an uncountbaly infinite set

R={x: x is a real number}

or

[0,1]={x: x is a real number and 0<=x<=1}.

Both of these are uncountably infinite sets.

There is no such thing as a 'potentially infinite set' in math. Sets are either finite or infinite. In the latter case, they are actually infinite. They can be countably infinite or uncountably infinite. For the uncountbaly infinite sets, there are infinitely many different possible cardinalities (although very few are used in practice)

Look in *any* math book and you will find NO distinction made between potential and actual infinity. The reason? Those notions have been replaced. They are no longer used because they are confused and ill defined.

So, I will make a challenge. Look in *any* advanced level math book produced in the last half century. Find *any* reference that discusses *at all* the notion of 'potential infinity'. I challenge you to find a single source *in math* for your claims from the last 50 years (I'll even go 75 years).

Okay, I was wrong about what mathematicians consider their sets to be. They do consider them to be actual infinite. But what does that mean? It does not mean the same thing a physicist (or even a philosopher) would mean with the same phrase. 

However, my point was better said in my post from page 6:

Quote:In the philosophy of mathematics, the abstraction [defined below] of actual infinity involves the acceptance (if the axiom of infinity [LINKED TO BELOW] is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and each individual result is finite and is achieved in a finite number of steps. https://en.wikipedia.org/wiki/Actual_infinity {emphasis added}

Quote:In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1] https://en.wikipedia.org/wiki/Axiom_of_infinity

In fact, regarding the Axiom of Infinity article, later on the article says: 

Quote:Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation...

I am not claiming the merits of one set of axioms or another--only illustrating that axioms are assumptions and are not self-evident truths (more below on this). 

Quote:Abstraction in mathematics is the process of extracting the underlying essence of a mathematical concept, removing any dependence on real world objects  with which it might originally have been connected, and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena.[1][2][3] https://en.wikipedia.org/wiki/Abstraction_(mathematics) {emphasis added}

So, putting braces {} around 1, 2, 3, 4, 5... makes a new mathematical object: A set. This is not a thing found in the real world that they needed to describe. This is an abstract object  created by axiom. More specifically, a non-logical axiom: 

Quote:As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), often shown in symbolic form, while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.

In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom, or any mathematical statement, to be "true" is an open question...in the philosophy of mathematics.[5] https://en.wikipedia.org/wiki/Axiom#Mathematical_logic {emphasis added}

Further, when mathematicians say "exist", they do not mean the same thing as nonmathematicians:

Quote:...The use of the word "exist" is merely a grammatical convenience here; mathematicians and nonmathematicians do not mean quite the same thing by this word. Unfortunately, we mathematicians don't have a better word; to be more precise we would have to replace this one word with entire paragraphs. If we assume the Axiom of Choice, we are not really stating that we believe in the physical "existence" of those sets or functions. Rather, we are stating that (at least for the moment) we will agree to the convention that we are permitted write proofs in a style as though those sets or functions exist.

Whether those sets or functions "really" exist is actually not important, so long as they do not give rise to contradictions. Mathematicians are perfectly willing to use devices that may be fictional, as intermediate steps in getting from a real problem to a real solution. https://math.vanderbilt.edu/schectex/cou...ntial.html

This last one is actually an easy to read article that explains things in plain language. 

THEREFORE, because infinite sets are a results of non-logical axioms (assumptions) that are not self-evidently true AND, the term 'exist' is not the same in mathematics, I stand by my claim that infinite sets in mathematics are not an indication that an actual infinite series or objects can exists in reality.

But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.

I'd point out that *finite* sets in mathematics are just as 'abstract'. mathematics is the study of formal axiom systems. So of course it is all abstract.

The idea that axioms are 'intuitively obvious' is another outdated idea. Euclid tried that with Euclidean geometry, but we have found that his system, even of geometry, was far from unique. Non-euclidean geometry is equally consistent, but gives different answers.

Which leads to the point: when there are different axioms systems that are all consistent, there is nothing to say which is correct and which is not correct except to go to observation and testing.

So, the fact that introducing actual infinities does bring a contradiction shows that there is no *logical* reason to exclude actual infinities.

And I agree--the actual existence of actual infinites has not been proven. But that isn't my claim. My claim is that there is no *logical* issue with them and that they should be considered as one *possibility*. And that is quite enough to destroy the Kalam argument. There is no *contradiction* with having an infinite regress of causes. It is internally consistent and so cannot be dismissed out of hand.

Nonlogical axioms? Way to diss mathematics, Steve. You will say anything for God to make sense. Interestingly enough, Cantor was a believer and was strengthened in his faith because of actual infinities.

Edit: Just read the quotes in full. Nonlogical meaning different from what I initially thought. Still, no logical contradictions

(February 26, 2018 at 2:55 pm)polymath257 Wrote:

(February 26, 2018 at 2:43 pm)SteveII Wrote: THEREFORE, because infinite sets are a results of non-logical axioms (assumptions) that are not self-evidently true AND, the term 'exist' is not the same in mathematics, I stand by my claim that infinite sets in mathematics are not an indication that an actual infinite series or objects can exists in reality.

But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.

The math is based on axioms (assumptions). It is question begging (circular reasoning) to say it is proof that they are logical possibilities. You have assumed an actual infinite in order to do further math with it. So, it gives no help to the argument that an actual infinity can exists. Therefore need we turn to something other than math:

1. You cannot get to infinity by successive addition.
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

These coupled with the fact that we don't have anything in the real world that could be an actual infinite leads a rational person to the believe that an actual infinite of real objects is not possible.

(February 26, 2018 at 6:43 pm)SteveII Wrote:

(February 26, 2018 at 2:55 pm)polymath257 Wrote: But the math shows that such assumptions lead to no internal contradictions: they are logical possibilities. And that is the whole point: that there is no *logical* obstacle to these being real.

The math is based on axioms (assumptions). It is question begging (circular reasoning) to say it is proof that they are logical possibilities. You have assumed an actual infinite in order to do further math with it. So, it gives no help to the argument that an actual infinity can exists. Therefore need we turn to something other than math:

1. You cannot get to infinity by successive addition.
2. You get absurdities when you propose an infinite number of actual objects (Hilbert's Hotel).
3. You get contradictions about how many squares and square roots there must be (Galileo's paradox)
4. Is the vase full or empty in the Ross–Littlewood paradox?
5. Is the lamp on or off in the Thomson's lamp paradox?
6. It seems we cannot traverse even a finite distance in Zeno's paradoxes

These coupled with the fact that we don't have anything in the real world that could be an actual infinite leads a rational person to the believe that an actual infinite of real objects is not possible.

On the contrary, the fact that we do not get inconsistencies from these axioms shows there is no logical problem with them.

We know of many axiom systems that *are* self-contradictory. The theory of infinite sets is not one of them.

1. Irrelevant. That isn't the mechanism for getting infinite sets.
2. Not absurdities, just counter to intuition derived from the study of finite things.
3. Not a contradiction. Again, just counter to intuitions derived from the study of finite things.
4. Not well defined. An impossible task due to relativity.
5. Not well defined. Task impossible to do because of relativity.
6. Resolved because both space and time are infinitely divisible: see algebra and calculus.

the lack of coherent arguments against actual infinities and the fact that they are not self-contradictory is enough to show they are *possibilities*.

Again, whether they are present in the real world is not known. But there is no *logical* issue. Paradoxes because people think in terms of finite sets, yes. But no contradictions.

---

Once again, Steve, you cannot get Graham's number (which is finite) via successive additions for any operations, as fast as they could be, in the current age of the universe.

Do you think Graham's number can exist as a logical possibility?

I dont know about some of the other paradoxes as Im not too familiar with them, but regarding Zenos continuity paradoxes, supposing it is a problem for motion, this universe might be both finite and discrete anyway ... maybe. Still doesnt mean infinite divisibility is illogical. But Zeno also had paradoxes that seem to show that motion was impossible even with discrete objects. Does this mean motion is illogical, Steve?

Also Grahams number is a beast. Youll find it fascinating when you read up on it. Fun read.

Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

(February 27, 2018 at 12:35 am)Jenny A Wrote: Steve,

Sorry to be late in the conversation.  But I really am puzzled as to why you think ifinities pose a logical contradiction as opposed to being hard to show in reality. 

I think the hotel idea sounds intuitively wrong to you merely because it's mathematical.   Infinite odd and infinite even numbers may feel wrong but isn't.  Think of it without numbers.  Suppose there is an infinite number of smoking rooms and an infinite number of non smoking rooms.  Two infinities of different kinds of things.  Together, both are infinite, yet you have twice as many room choices should you  be ambivalent to smoke.  

If that's still too close, consider infinite cats and infinite dogs .  Both infinities together are infinite pets.  Still infinity but with more choices.  

No logical contradiction.  Just more kinds in the omega.

Yep both are without ends so naturally the total set will be without end as well.