An infinite past?

_{kairosfocus

January 26, 2016

Atheism, Logic and Reason, Mathematics, Sciences and Theology, worldview}

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In the current UD thread on Darwinism and an infinite past, there has been an exchange on Spitzer’s argument that it is impossible to traverse an infinite past to arrive at the present.

Let me share and headline what is in effect the current state of play:

DS, 108: >>KF,

DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out.

There are oscillating universe models which are consistent with an infinite past, as I stated. Replace each tick with a big bang/crunch cycle.

And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown.

No. I am saying that Spitzer assumes that an infinite number of ticks cannot in principle occur as part of his argument. The burden is on him to prove that.

A down to 2, 1, 0. Where A = 1/m, m –> 0 i.e. is infinitesimal. KF

There are no infinitesimals in sight in your statement above. All the numbers are real and finite.

Now that we’ve clarified the ticking vs. counting issue, do you still have mathematical (not physical) objections to the eternal ticking clock example? If so, I challenge you to take some time and write them out precisely (avoiding such concepts as “of order aleph-null” and abuse of the hyperreals). Maybe even post it as an OP.>>

To this I responded:

KF, 112:>>Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges.

Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points.

Beyond, Spitzer summarised arguments that the transfinite cannot be traversed in separate finite steps. He did not merely assume.

Above, the exchange we have had is about precisely that.

You have admitted that you are unable to show such a traverse, and are now adverting to oscillating models that have failed.

I have taken time step by step to put the challenge in terms of completing the arrival at the present; in the face of many objections on your part.

You have spoken of how at any specific point, already an infinite number of steps is complete. I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by (in the end) using some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes. The focal task then becomes traversing onward from A to 0, envisioned for the moment as the singularity, from which onward we go to now, n. Where as you objected to negatives [though how that was used was explained] I use asterisks to show the finite up-count since the singularity. Of course the lead ellipsis indicates that A is not the beginning of the steps we may identify and list as a succession, it is preceded by an arbitrarily and per your suggestion for argument even possibly transfinitely large and unending set of previous values:

. . . A, . . . 2, 1, 0, 1*, 2*, . . . n*

Such, of course was already outlined by way of making the way clear after successive objections.

The start point for a count is arbitrary, so let us put the start at A and put it into correspondence with the naturals, i.e. this is in principle countable . . . as is implicit in stepwise succession as would happen with clock ticks, one providing the basis for the next as energy is gated from a source and as positive, precisely lagged feedback is applied:

A, (A less1), . . .
0, 1, . . .

Given that the traverse from A to 0 is transfinite, the task here is comparable to counting up from 0 to a transfinite in finite successive steps, which is a supertask that is unattainable. (And I have taken the step of identifying A as a specific number a reciprocal of a number close to 0 [as the hyper reals approach takes to identify what an infinitesimal is, only in reverse], to avoid all sorts of issues on what does subtraction mean with a transfinite. Such will of course be of at least the order — scale if you will — of aleph-null from the origin at 0. I take it that we can accept the reasonableness of infinitesimals close to but not quite attaining to zero; such being foundational to a way to understand the Calculus.)

For, once we count 0, 1, . . . n, we may always go on to n +1, etc in further steps, always being finite.

The evidence is that traversing an infinite succession of finite discrete steps is a unattainable supertask, precisely as Spitzer sums up.

The worldviews significance is this, that a contingent succession of beings, with each being b_i subject to on/off enabling causal factors it must have in place for it to begin or continue to exist, must be a part of a chain of successive and in context finite discrete causes. This can be in principle enumerated and compared to the step-wise succession, e.g. of clock ticks on a clock.

We then see that the traversal of an infinite succession of such beings is to be doubted, on grounds of needing to arrive at the singularity then onward up to us. From the singularity (for reference to current cosmology, actually any reasonable zero point would do equally) to us is explicable on a succession, but the problem is to arrive at 0.

This may then be multiplied by the challenge that non-being, the genuine nothing, can have no causal powers. There is not space, time-point, energy, mass, arrangement, mind etc “there.” So were there ever utter nothing, such would forever obtain.

We face then, the need for a necessary root of being to account for a world that now is.

Necessary, so connected to the framework of a world that no world can be absent such. As an instance, 2 must exist in a world W where distinct identity, say A, exists: W = {A|~A}.

A world now is, so something always was.

Following, frankly, the line in the classic work, Rom 1:19 ff (which I find to be enormously suggestive of a frame for a reasonable faith worldview), this world is a world in which we find ourselves as self aware, responsibly free and rational individuals; contingent beings subject to moral government and intuitively sensing the need to respond appropriately to evident truth about ourselves and our circumstances in a going concern world.

It is appropriate in such a context to ask, what sort of serious candidates — flying spaghetti monsters etc are patently contingent imaginary parodies that do not meet the criteria for necessary being and need not apply — can we see in making a worldview level choice?

After centuries of debate, there is one serious candidate, by utter contrast with non-serious parodies, and by contrast with the challenge of traversing the transfinite etc.

The bill to be filled looks extraordinarily like:

an inherently good creator God, a necessary and maximally great being worthy of loyalty and the responsible, reasonable service of doing the good in accord with our nature.

This is Candidate A.

Candidate B is: ___________ ?>>

It seems to me, that this is the underlying worldview level issue, and that as usual, the question pivots on just what is it that can be seen in comparing difficulties of start points, first plausibles. And of course as every tub must stand on its own bottom, DS is just as duty-bound to show why he thinks an infinite successive finite step traverse is impossible as he thinks Spitzer is to justify his assertion that such is not possible. And in the bargain, I think I have stipulated that m is infinitesimal and have taken reciprocal 1/m = A as a transfinite, specific whole number in reverse of the general approach used in defining hyper reals and using the concept that properties of reals extend in the argument to the hypers. END

Comments

Phineas, whether you try to count down from a transfinitely remote point or up to it makes little difference. A transfinite span cannot be bridged in discrete finite steps. For going up, count 0, 1, 2, . . . n. At any n arrived at the result is inherently finite and can go on to the next finite n + 1 and so on. You will never reach the transfinite stepwise. A is not a beginning, cf the OP, it is just a convenient but transfinitely remote point along the way to start counting to show the problem of bridging to 0 from an infinitely remote causal chain and temporal past. KF PS: I clip the OP:
You have spoken of how at any specific point, already an infinite number of steps is complete. I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by (in the end) using some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes. The focal task then becomes traversing onward from A to 0, envisioned for the moment as the singularity, from which onward we go to now, n. Where as you objected to negatives [though how that was used was explained] I use asterisks to show the finite up-count since the singularity. Of course the lead ellipsis indicates that A is not the beginning of the steps we may identify and list as a succession, it is preceded by an arbitrarily and per your suggestion for argument even possibly transfinitely large and unending set of previous values: . . . A, . . . 2, 1, 0, 1*, 2*, . . . n* Such, of course was already outlined by way of making the way clear after successive objections. The start point for a count is arbitrary, so let us put the start at A and put it into correspondence with the naturals, i.e. this is in principle countable . . . as is implicit in stepwise succession as would happen with clock ticks, one providing the basis for the next as energy is gated from a source and as positive, precisely lagged feedback is applied: A, (A less1), . . . 0, 1, . . . Given that the traverse from A to 0 is transfinite, the task here is comparable to counting up from 0 to a transfinite in finite successive steps, which is a supertask that is unattainable. (And I have taken the step of identifying A as a specific number a reciprocal of a number close to 0 [as the hyper reals approach takes to identify what an infinitesimal is, only in reverse], to avoid all sorts of issues on what does subtraction mean with a transfinite. Such will of course be of at least the order — scale if you will — of aleph-null from the origin at 0. I take it that we can accept the reasonableness of infinitesimals close to but not quite attaining to zero; such being foundational to a way to understand the Calculus.) For, once we count 0, 1, . . . n, we may always go on to n +1, etc in further steps, always being finite. The evidence is that traversing an infinite succession of finite discrete steps is a unattainable supertask, precisely as Spitzer sums up . . .
kairosfocus_{January 26, 2016
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DS, please. You know better, far better. If there are infinitely many naturals, there will be no last member, hence the ellipsis. If you look I built that in to the down count challenge, you are setting a start count from A which is itself not the actual first, but is a convenient start for a count. Then the problem is it is transfinitely remote from 0 and a count process will always only reach a finite number. In short the transfinite range cannot be bridged to get to the big bang singularity on the assumption time has an infinite past. And you obviously have not reckoned with why A is a transfinite integer. That allows us to set a start point for the down count at a transfinitely remote INTEGER, and contrary to your it's like a square circle stunt, they exist. Likewise a set 1, 2 has span 2, 1, 2 . . . 10^500 has 10^500, The full set is infinite and so must embrace some transfinite A etc. KFkairosfocus_{January 26, 2016
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KF,
DS, how many natural numbers are there, what is the biggest and last one and how do you distinguish listing in succession from a count. Last I checked N is subset of Z and onwards R thence C.
There are infinitely many natural numbers. The set has cardinality aleph-null. There is no biggest or last one. Do you agree? I'm asking for a direct yes or no answer here, because I sincerely can't tell, based on your above posts. To me,
If ALL of its members so enumerated or indicated in succession by ellipsis are finite, the set cannot but be finite.
means that any set consisting of (say) finite integer values must be finite. Is that not what you meant? When I "count" a set S, I am usually referring to setting up a correspondence between some subset of the natural numbers and S. Listing in succession does not imply knowledge of a correspondence. And yes, I agree with the chain of set inclusions that you described.
DS, to identify that in principle a real number can be expressed in a PVN form does not imply that such will be exhaustively possible any more than can counting count all naturals.
Surely you are aware of the fact that the reals are in one-to-one correspondence with non-terminating decimal expressions (and a + or - sign)? No non-real hyperreals can be expressed in that form.
PS: You have not engaged the definitions of a hyperinteger given this morning in 4 above. Those are crucial.
They aren't crucial, because I stipulated that the clock ticks occur only at (real) integer time coordinates. No hyperreals involved here. If you produce an infinite hyperreal A, then a tick did not occur at time -A seconds.daveS_{January 26, 2016
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I'm probably way in over my head, but is counting down from infinity past to now the same thing as counting from now back to infinity past? I see the problem with the former, but not so readily with the latter. Edit: To a certain extent, framing the issue as counting down from infinity past to now would seem to assume the very same starting point (named as infinity past) that it is trying to prove. To my quite possibly under-educated mind in any case. Further Edit: Put another way, the idea of using infinity past as a starting point seems like a non-starter (if you will). If you can't reach it from here, then you can't start there. Similarly, you couldn't use infinity future as a starting point. The starting point can only be either the present or a finite number of steps from the present. Final Edit (I promise): It seems like the argument is being framed as: Assume a beginning a finite number of seconds ago vs Assume a beginning an infinite number of seconds ago But I think what is being argued is: Assume a beginning a finite number of seconds ago vs Assume no beginningPhinehas_{January 26, 2016
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DS, to identify that in principle a real number can be expressed in a PVN form does not imply that such will be exhaustively possible any more than can counting count all naturals. We cannot fully write pi or e, but that does not mean they are not real. Nor does this answer to what the open ended ellipsis entails for listing N. And so forth. At this point, it looks like you are flailing rather than addressing your core challenge, to show how one gets to just now in effect count down from infinity past to now. I suggest this is an infeasible supertask and time cannot be infinite in the past. I add, nor can a chain of successive causal events such as clock ticks or beings such as would chain back from the present of our world to the singularity and to whatever is beyond such. And yes this cries out for necessary being of eternity to be acknowledged as real and the root in which time comes to be. KF PS: You have not engaged the definitions of a hyperinteger given this morning in 4 above. Those are crucial.kairosfocus_{January 26, 2016
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DS, how many natural numbers are there, what is the biggest and last one and how do you distinguish listing in succession from a count. Last I checked N is subset of Z and onwards R thence C. KFkairosfocus_{January 26, 2016
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KF,
DS, if a set that effectively counts from 0 or 1 in succession has in it transfinitely many members in aggregate, it must span to transfinite numbers, or equivalently span to infinity.
What does it mean for a set to "span to infinity"? There have been so many misunderstandings in this discussion that I want to be sure we're speaking the same language. Does the set of natural numbers (which are all real) "span to infinity"?
If ALL of its members so enumerated or indicated in succession by ellipsis are finite, the set cannot but be finite.
Do you therefore claim that any set which has only finite integer members is therefore finite? This is absolutely false, if that's what you're saying.
Beyond such, with all due respect some of your questions are now trivial to the point of being questionable, if we both agree that we understand the standard place value notation.
My point in asking that question was the following: expressions of the form:
1 * 10^1 + 9 * 10^0 + 7 * 10^-1 + 8*10^-2 + 0 *10^-3 . . .
will always be real, if your b's, n's and d's are real (and presumably b > 1). You will not obtain any nonreal hyperreal numbers in that manner.daveS_{January 26, 2016
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DS, if a set that effectively counts (i.e. successively lists) from 0 or 1 in succession has in it transfinitely many members in aggregate, it must span to transfinite numbers, or equivalently span to infinity thus including members of transfinite scale. If ALL of its members so enumerated or indicated in succession by ellipsis are finite, the set cannot but be finite. Indeed, the ellipsis with an open end indicates, continued without any finite uppermost term, we can only point there. Beyond such, with all due respect some of your questions are now trivial to the point of being questionable, if we both agree that we understand the standard place value notation. Your earlier remarks left me in doubt in this regard. KFkairosfocus_{January 26, 2016
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I am sorry, but I am trapped in the infinite now and I hear only one clock tick.Mung_{January 26, 2016
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KF,
DS, sorry but a set of integers that spans to infinity will have members that are transfinite.
I have to believe we are not understanding each other here. The set of natural numbers (say) has cardinality aleph-null but has no non-finite elements. Do you disagree with this? [Edit: I'm interpreting "spans to infinity" in the obvious manner, I believe, although I wouldn't use that phrase].
PS: Do you recognise place value notation and the power series it represents? Such is a commonplace of computing and digital electronics as well as mathematics.
Of course. I'm asking you to tell me what kind of numbers these are. Rationals, integers, something else?daveS_{January 26, 2016
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DS, sorry but a set of integers that spans to infinity will have members that are transfinite. We cannot count to such members or count down from such (what the ellipsis in part indicates), or even write them down but we can point to and construct such on reasonable terms. That is how I come to highlight what I symbolised as A. KF PS: Do you recognise place value notation and the power series it represents? Such is a commonplace of computing and digital electronics as well as mathematics. I used b for base as stated. d is the corresponding set of digits, commonly decimal, duodecimal, binary, hexadecimal and sexagesimal. n is the power of the base corresponding to places, in primary school, . . . hundreds, tens, units, tenths etc, in higher levels relevant powers of the base.kairosfocus_{January 26, 2016
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KF, I think the hyperreals will end up being not relevant but in this expression:
SUM [d*b^n]: e.g. 19.78 = 1 * 10^1 + 9 * 10^0 + 7 * 10^-1 + 8*10^-2 + 0 *10^-3 . . .
From what set are the b, n, and d drawn from?daveS_{January 26, 2016
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KF,
DS, are you saying your clock has only been ticking for finite time? If not, it has to have been ticking a transfinite number of ticks past, and so would have been ticking at A if such were possible. KF
It has not been ticking for only a finite time. The time coordinates (in seconds) at which the clock has ticked before present comprise the set {-1, -2, -3, ... } (the negative integers). The set is infinite, but none of its members is transfinite.daveS_{January 26, 2016
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F/N: The significance of this issue is, if a space-time plenum is all there has been, there must have been a past temporal causal succession. Where, either it is finite and something has come causally from non being at a start point or else a transfinite span of causes has been traversed in succession. Neither of these is attractive as a worldview option, indeed there are questions of incoherence or absurdity. The pivot is, why is there something rather than nothing? KFkairosfocus_{January 26, 2016
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DS, are you saying your clock has only been ticking for finite time? If not, it has to have been ticking a transfinite number of ticks past, and so would have been ticking at A if such were possible. KF PS: I believe there is an answer already given, in the context of hyperintegers if you will, cf 4 on such -- whole, transfinite numbers. Note the OP: . . . A, (A less 1), . . . 2, 1, 0, 1*, 2* . . . n, n now. A = 1/m. m an infinitesimal whose reciprocal will have fractional part F = 0000 . . .kairosfocus_{January 26, 2016
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KF, I'll look at your post #6 more closely later, but I'll say it again: my example does not involve counting down from A to 0, for some transfinite A. Edit: I see you referred to that above. It remains true that there is no clock tick a transfinite A seconds before present. If you think there is, identify its time coordinate in this set: {..., -2, -1, 0}.daveS_{January 26, 2016
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DS, Kindly note Keisler, and the summary in Wiki that derives therefrom. I believe it is fair summary that a real (or extensions thereof) can be represented with a place value, whole + fraction notation that is a shorthand for a power series on a base with each term b^n multiplied by a digit, d: SUM [d*b^n]: e.g. 19.78 = 1 * 10^1 + 9 * 10^0 + 7 * 10^-1 + 8*10^-2 + 0 *10^-3 . . . In that context we use N = W.F, and of F is 0000 . . . then it is reasonable to speak of a whole. Or integer. Thence by extension hyperinteger. All I am doing is taking that infinitesimals exist close to 0 and their reciprocals are hyper-reals. I take some m of that order and stipulate its reciprocal A has a whole part with F = 0000 . . . Then, start counting down from A to 0. I add i/l/o your comment, for emphasis, that A is not the first tick if you will, it is the one where the count picks up, only it is at a transfinite distance on the number line from 0. Never mind your mathematical clock was ticking before that and per argument arbitrarily long before that, the issue now is to count from A down to 0. Thus can be set in correspondence with 0. 1, 2, / . . and will run into the problem of stepwise traversal of a transfinite span. KFkairosfocus_{January 26, 2016
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KF,
DS, I suggest to you there is no more inherent reason why a number with no fractional part should not be transfinite than that one with such a part should not be.
This has nothing to do with fractional parts. The standard definition of the set of whole numbers is {0, 1, 2, ...} (possibly without 0), none of which are transfinite. Each whole number has a decimal representation. If I ask you to name a transfinite whole number via decimal representation, you won't be able to.
Seems reasonable to me. I would infer from that that I am dealing with a pair of reciprocals, A and m, such that 1/m = A, and where A is an integer of transfinite character. If you can show different, kindly do so.
I have no issues with this. But then I'm not clear why you spoke of letting some number m (real, hyperreal, I don't know) approach 0 to arrive at an "infinitesimal", rather than simply choosing a single hyperreal infinitesimal directly.
But the point remains, that you need to show us that we can descend to 0 step by finite discrete step from a transfinite.
As I've stated before, that doesn't occur in my example. See below.
Whether climbing up or down the ladder at any point we are only ever at some n, a finite distance from the point we started.
But we/the clock didn't start, as my post #76 explains in the other thread. It is eternal, without beginning (but it could have an end without changing anything). The ticks occur at these times in our coordinate system: {..., -2, -1, 0, 1, 2, ...} There is no tick A seconds before the present if A is a transfinite number. Therefore there is no traversal A, A - 1, ..., 2, 1, 0.daveS_{January 26, 2016
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DS, I suggest to you there is no more inherent reason why a number with no fractional part should not be transfinite than that one with such a part should not be. After all, a whole number is effectively one where the fractional part is F = .00000 . . . And, given that there is still a matter of discrete causal succession at stake, the matter in question comes inherently in discrete states which can be assigned integer values. For convenience, Wiki:
In non-standard analysis, a hyperinteger N is a hyperreal number equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence (1,2,3,...) in the ultrapower construction of the hyperreals.
U/D: Keisler, p 159:
We begin with a new concept, that of a hyperinteger. The hyperintegers are to the integers as the hyperreal numbers are to the real numbers. The hyperintegers consist of the ordinary finite integers, the positive infinite hyperintegers, and the negative infinite hyperintegers. The hyperintegers have the same algebraic properties as the integers and are spaced one apart all along the hyperreal-line
Seems reasonable to me. I would infer from that that I am dealing with a pair of reciprocals, A and m, such that 1/m = A, and where A is an integer of transfinite character. If you can show different, kindly do so. But the point remains, that you need to show us that we can descend to 0 step by finite discrete step from a transfinite. That is for sure what you have to mean by suggesting that at any given point there was already an infinite succession completed to that point. It is that assumption of infinite succession you and others have made that needs justification, and I cannot see it having any for precisely the reason that we cannot ascend from 0 to the transfinite step by step either. Whether climbing up or down the ladder at any point we are only ever at some n, a finite distance from the point we started. Conventionally, we start at 0. But we obviously need not do so. For . . . -2, 1, 0, 1, 2 . . . --> do we not start elsewhere to add? 2 + 3 => 2 -> 3, 4, 5| So, why not start at transfinite A = 1/m, m an infinitesimal close to 0, and descend? If not, why not? Just what contradiction of core characteristics is there, as happens with a square circle? If so, how can one traverse the descending range to 0 step by step? KFkairosfocus_{January 26, 2016
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KF, As I mentioned in the other thread, you need to remove all references to "transfinite whole numbers" from your argument. They don't exist, just as square circles don't exist. Edit: Ok, I need to qualify that: I'm taking "whole numbers" here to comprise the set {0, 1, 2, ... }, as defined by the Wolfram site.daveS_{January 26, 2016
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PS: Observe how h functions here and especially the implications of h --> 0: https://diversity.umn.edu/multicultural/sites/diversity.umn.edu.multicultural/files/DifferentiationRules.pdf and compare the discussion per non standard analysis here: http://www.math.wisc.edu/~keisler/calc.htmlkairosfocus_{January 26, 2016
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Is an infinite past traversible in a causal succession of distinct, finite duration steps? Spitzer argues no, DS doubts his argument, I suggest Spitzer has a serious point.kairosfocus_{January 26, 2016
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