In this forum, someone asked:
>>I have been wondering if a^0 is DEFINED to be 1 or is PROVED to be 1.<<

The final answer to the question was,

>>So ultimately, which of you is right depends on what axiomatic system you start with, and how you define exponents initially.<<

So, in my way of seeing things, this is something artificial, because you have to define it to be 1, and you define it to be 1 just to keep the formulas. You can disagree, if you start with a different definition of exponents.
The reason for our disagreements about probabilities probably is the same.


If I define the theory of probability as the study of randomness, and if an event is a certainty, it’s artificial to speak of it in terms of probabilities.


The meaning of what I wrote is clear by the example I gave:

>>I would never say that Mary’s chance of getting pregnant without the participation of a male was of 100%, even if God had revealed to me in a dream that she would get pregnant. I would say that her chance of getting pregnant without the participation of a male was zero – that’s why what happened was a miracle.<<

I was speaking of the prior probability, while you were speaking of the posterior probability.


I agree with this argument. What I was trying to point out during the whole discussion is that the fact that there is no risk that something will not occur doesn’t mean there is no risk posed to you. What I mean is that a risk is posed by a threat, that is, by something negative that can happen to you.
Of course there is no risk that your death will not occur – this is a certain event. But the risk to you is death itself – this is a risk in terms of something negative that can happen to you.

It's what you're not saying that is unclear. Do you believe the Bible clearly says the Father feared Jesus would fail, and sent Him anyhow hoping He would succeed? If so, please post the biblical quotes which clearly say so. If you can establish this point, then we can study what Ellen meant when she used the word "risk" in relation to Jesus' messianic mission.

What do you think Ellen White meant when she employed the word "risk" in relation to Jesus' messianic mission? Do you think it required the Father sending the Son not knowing in advance if Jesus would fail or succeed?

No, in my view it doesn't require that. As I explained some pages ago, the word "risk" is applied to Christ, not to God. God sent His Son to meet a risk.

Here is a sampling of how Ellen White employed the concept of risk:

He permitted Him to meet life's peril in common with every human soul, to fight the battle as every child of humanity must fight it, at the risk of failure and eternal loss. {DA 49.1}

To meet a bitterer conflict and a more fearful risk, God gave His only-begotten Son, that the path of life might be made sure for our little ones. {DA 49.2}

Then as the glories of the eternal home burst upon our enraptured senses we shall remember that Jesus left all this for us, that He not only became an exile from the heavenly courts, but for us took the risk of failure and eternal loss. {DA 131.2}

Satan wished to cause Jesus to presume upon the mercy of His Father and risk His life before the fulfillment of His mission. He had hoped that the plan of salvation would fail; but the plan was laid too deep to be overthrown or marred by Satan. {EW 156.1}

But in the heavenly race we can all run, and all receive the prize. There is no uncertainty, no risk, in the matter. {CH 47.1}

But in complying with the heavenly invitations we have no such risk to run. We must take God at His word, and in simplicity of faith walk out upon the promise, and give to the Lord His own. {CS 90.1}

Those who defer obedience till every uncertainty disappears, and there remains no risk of failure or defeat, will never obey. {GW 262.1}

Ezra would run the risk of trusting his cause with God. He well knew that if they failed in their important work, it would be because they had not complied with the requirements of God and therefore He could not help them. {3BC 1134.8}

In order to do this he has had to take responsibilities and to risk something on the success of this message. God would be pleased if others would feel the same interest, and move with the same energy, but many will not venture. {2SG 280.3}

I would recommend letting them go without for at least three days until they are hungry enough to enjoy good wholesome food. I would risk their starving. {Te 158.2}

In that important position God will have a man to venture, to risk something; to move out firmly for the right, whatever may be the consequences; to battle against obstacles, and waver not, even though life be at stake. {1T 320.2}

Some have no idea of running any risk or venturing anything themselves. But somebody must venture; someone must run risks in this cause. {3T 316.1}

They must be willing to run some risk, to venture something. It is not pleasing to God that we defer present opportunities for doing good, in hope of accomplishing a greater work in the future. {5T 392.1}

There is a prospect before us of a continued struggle, at the risk of imprisonment, loss of property, and even of life itself, to defend the law of God, which is made void by the laws of men. {5T 712.3}

It was their own willful unbelief that turned them back. They were unwilling to risk anything upon the promises of God. {TMK 169.4}

So, knowing He would succeed didn't lessen the risk of failure and eternal loss Jesus faced? Did Jesus also believe He would succeed? If so, did it lessen the risks He faced? If not, did He ever doubt He would succeed?

Do you think 8/8=1 is artificial?
Or proofs in geometry? Something similar comes to mind of showing congruent angles using the same sort of logic as Tom did for exponents. Now, maybe what you are really saying is artificial is the subtraction of exponents. If so, we could discuss that. But saying 2^0 = 1 is artificial, I can't go with that or would have to discount all I learned in geometry proofs.

Have you taken a probability and statistics class?
I perceive you have a different definition of probability.

I wasn't talking about Saul, and not sure why you thought that. I was talking about how the Bible may not mean what it appears to say and requesting you support your similar assertion regarding Ellen White. Please read Tom's post regarding your comment. He said it well.


Could you explain what you mean by that statement?
Do you see risk relating to uncertainty? If not, what do you think "risk" means?

I had already responded regarding your request. Even if I could meet your request, since you appear to have some different definition of what "risk" means, it would be foundational for you to explain what you mean by the above statement. Otherwise, how could someone possibly "post a plain, Thus saith the Lord" regarding risk if you don't define what you mean? I assume that's not your point.

So how do you see peril in common with every human soul not being risk? Guess you haven't defined it yet.

It's great that this sort of thing is being discussed on this forum!

I think what was said is incorrect. The axioms which are the basis of Mathematics are set theory axioms. There is some disagreement in regards to how infinities should be handled, but I don't believe there are any disagreements which would involve how exponentiation should be handled. There's only one way, and there should be a number of ways of proving this; Alegebra, Real Analysis, and Complex Analysis should all lead to the same definition.



This is coming at things backwards. First you start with the set axioms, and then you construct things (groups, and fields). You construct the positive integers, then natural, then all integers, then rational numbers, then irrational numbers, transcendental numbers, the real number field, and finally complex numbers and the complex number field. In the process of construction certain operations are defined, and there's no scope for defining these operations differently in the construction of these groups/fields.

So it's not an artificial process, but one which follows logically from the set theory axioms from which one starts. The statement that:



makes sense to me hypothetically, but the axiomatic system which is the basis of Mathematics are the set theory axioms, and while there is some disagreement in regards to how infinities should be handled, I'm not aware of any axiomatic structure which would impact exponentiation. If there is such an axiomatic structure, I'd be interested in hearing about it.



It sounds like there is some confusion here in regards to thinking that there is an artificial definition involved, when there isn't, for both exponentiation and probabilities. So I would agree that our disagreements are probably the same, in that you see things being done artificially when there's really nothing artificial about it.



"Probability" is



The numerical chance of a specific outcome occurring is a number between 0 and 1 inclusive. There is nothing artificial about this definition.



I don't see that what you're saying makes any sense. Specifically



If God revealed to you in a dream that she would get pregnant without the participation of a male, then if you maintained that she had no chance of becoming pregnant without the participation of a male, then you would be disagreeing with what God revealed to you. You would be simply not believing God, and you would be wrong in your assertion that there was no chance of Mary getting pregnant.



Glad to hear that!



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I have no idea what point you're wishing to make here. Perhaps you could give a concrete example.

I'll give one using the argument you said you agreed with.

1.If God is certain an event will occur, say X, then X is certain to occur.
2.If X is certain to occur, then there is no risk that it will not occur.

Say X is that Christ would not be overcome by any temptation. If God was certain that Christ would not be overcome by any temptation, then it is certain that Christ would not be overcome by any temptation, and God incurred no risk in sending Him. I don't see anything to disagree with here, given you agreed with my argument presented above. This is just specifying a specific event for "X".

No, I think 2^0=1, or 1,000,000^0=1, is artificial. You have to define what it is. 2^2 is 2*2, but 2^0 is what?


If there weren't random events, what would you use the probability theory for?
I've googled the words probability "study of randomness" and obtained 58,200 results, and you say that I have a different definition of probability?