Questions about omnipotence paradox

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ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Felix wrote:My point is that it is not logical.
When I said that the omnipotence paradox is logical, I meant that it is related to logic (‘logical’ as the antonym of ‘non-logical’) and not that it is in accordance with the laws of logic (‘logical’ as the antonym of ‘illogical’). I assume that you too use the former sense in what you said above, i.e. that you mean that the omnipotence paradox is not related to logic. This means that you agree with me on the point that logic cannot be used to solve the omnipotence paradox. You justify this by saying that the paradox is not logical in the first place.
Felix wrote:Logic can only evaluate definite terms.
This is the reason you give for the paradox not being logical; because the omnipotence paradox contains an indefinite term, omnipotence, and logic can only evaluate definite terms, the paradox is beyond logic because it deals with concepts that are not within the bounds of logic.

I agree that the paradox is beyond logic (this has been my preferred stance all along). I also like the reason you give for that (that it contains terms beyond logic). The arguments Arg1, Arg2 and Arg3 also demonstrate that logic cannot deal with omnipotence.

Felix wrote:"Omnipotent Being" is itself a logical contradiction, because to be a being is a limitation on omnipotence.
This is something that I think should be realized much more widely. In fact, I find it inaccurate to describe God as the Supreme Being.

However, I do not see how that directly relates to the omnipotence paradox. The paradox does not make any mention of an omnipotent being. It only talks about God and implicitly assumes him to be omnipotent, but it does not claim that he is a being.


Have I understood you correctly?
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Whitedragon wrote:So should we not be asking whether infinity could be equal to itself?
Whether ∞ = ∞ is a very important question indeed regarding the omnipotence paradox. At first, it may seem to have a trivial, affirmative answer which follows directly from the fundamental law of logic that ∀ x : x = x. However, as Felix pointed out correctly in my opinion, normal logic cannot be applied to indefinite terms. Now, infinity is not definite. Therefore, we cannot use even the most fundamental laws of normal logic to "prove" ∞ = ∞, and your question thus turns out to be not that easy to answer after all.

Whitedragon wrote:but it is not the same sort of number as natural or real numbers. ... And since, neither infinity-infinity=0, nor infinity/infinity=1 you could say that infinity is not equal to infinity.
That might be a crucial realization in trying to solve the omnipotence paradox. Consequently, ∞ ≤ ∞ and ∞ > ∞ may be true at the same time. It follows that (strength of God) = ∞ ≤ ∞ = (weight of stone) (so God cannot lift the stone) and (strength of God) = ∞ > ∞ = (weight of stone) (so God can lift the stone) can be true simultaneously, possibly resolving the omnipotence paradox.


Is that the completed version of the solution you propose (just to make sure I have understood you correctly)?
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

By using Felix' realization that omnipotence is indefinite and therefore non-logical to start with and the extension of that realization to infinity in order to justify Whitedragon's claim that infinity could be unequal to infinity, which would normally be a simple violation of the laws of logic, we may have found a non-logical solution of the omnipotence paradox, just as I had hoped.

Still, since being questioning, critical and skeptical is one of the greatest virtues a philosopher can have, we should ask ourselves whether there might be problems with our possible solution.

Is it meaningful to talk of non-logical solutions in the first place?

Maybe, the realization that infinity can be unequal to itself is not an actual solution but just a statement that logic does not suffice to solve the omnipotence paradox, which we had already established in a different way with arguments Arg1, Arg2 and Arg3.

Is logic really limited to the definite? Transfinite numbers, for example, are not finite but still abide by the laws of logic. Furthermore, isn't the concept of universal quantification logical in spite of being infinite?
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Whitedragon
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Re: Questions about omnipotence paradox

Post by Whitedragon »

ChanceIsChange wrote:
Whitedragon wrote:So should we not be asking whether infinity could be equal to itself?
Whether ∞ = ∞ is a very important question indeed regarding the omnipotence paradox. At first, it may seem to have a trivial, affirmative answer which follows directly from the fundamental law of logic that ∀ x : x = x. However, as Felix pointed out correctly in my opinion, normal logic cannot be applied to indefinite terms. Now, infinity is not definite. Therefore, we cannot use even the most fundamental laws of normal logic to "prove" ∞ = ∞, and your question thus turns out to be not that easy to answer after all.

Whitedragon wrote:but it is not the same sort of number as natural or real numbers. ... And since, neither infinity-infinity=0, nor infinity/infinity=1 you could say that infinity is not equal to infinity.
That might be a crucial realization in trying to solve the omnipotence paradox. Consequently, ∞ ≤ ∞ and ∞ > ∞ may be true at the same time. It follows that (strength of God) = ∞ ≤ ∞ = (weight of stone) (so God cannot lift the stone) and (strength of God) = ∞ > ∞ = (weight of stone) (so God can lift the stone) can be true simultaneously, possibly resolving the omnipotence paradox.


Is that the completed version of the solution you propose (just to make sure I have understood you correctly)?
Okay, let us back up for one moment here; forget infinities until the next post. Let us ask another question, do stones exist in the spirit world? Secondly, if they do not exist in the spirit world; that means a deity would have to become a man and a 100kg would limit him easily, not even having to create a larger stone.
We are a frozen spirit; our thoughts a cloud of droplets; different oceans and ages brood inside – where spirit sublimates. To some our words, an acid rain, to some it is too pure, to some infectious, to some a cure.
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Whitedragon wrote:Let us ask another question, do stones exist in the spirit world?
Of what importance is that? Firstly, what does that have to do with the omnipotence paradox? Secondly, regardless of whether stones normally exist in the spirit world or not, God, being omnipotent, could create one in the spirit world. So, one version of the omnipotence paradox could run as follows:

Can God create a stone in the spirit world which is too heavy for him to lift?

Whitedragon wrote:Secondly, if they do not exist in the spirit world
This seems to imply that you believe God would have to create the stone in the spirit world, which might have been your answer to my question above as to how the spirit world relates to the omnipotence paradox. However, we would thus be presented with two problems. The first one is that God is omnipotent and therefore can create a stone of whatever property wherever he likes. The second one is that God's omnipotence would be limited by the laws of the spirit world, a violation of the definition of omnipotence.

Whitedragon wrote:that means a deity would have to become a man and a 100kg would limit him easily, not even having to create a larger stone.
If God is truly omnipotent, then that omnipotence is not limited to the spirit world but rather extends to everything and beyond.
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Whitedragon
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Re: Questions about omnipotence paradox

Post by Whitedragon »

ChanceIsChange wrote:
Whitedragon wrote:Let us ask another question, do stones exist in the spirit world?
Of what importance is that? Firstly, what does that have to do with the omnipotence paradox? Secondly, regardless of whether stones normally exist in the spirit world or not, God, being omnipotent, could create one in the spirit world. So, one version of the omnipotence paradox could run as follows:

Can God create a stone in the spirit world which is too heavy for him to lift?

Whitedragon wrote:Secondly, if they do not exist in the spirit world
This seems to imply that you believe God would have to create the stone in the spirit world, which might have been your answer to my question above as to how the spirit world relates to the omnipotence paradox. However, we would thus be presented with two problems. The first one is that God is omnipotent and therefore can create a stone of whatever property wherever he likes. The second one is that God's omnipotence would be limited by the laws of the spirit world, a violation of the definition of omnipotence.

Whitedragon wrote:that means a deity would have to become a man and a 100kg would limit him easily, not even having to create a larger stone.
If God is truly omnipotent, then that omnipotence is not limited to the spirit world but rather extends to everything and beyond.
Stones in the spirit realm do not make sense, because they would have to be ethereal. It is like asking can something be a gas and a solid at the same time, or the famous question; “can the Lord create a square circle?” To get back to infinities, some infinities are larger than others, but then they are not actual infinities. True infinity cannot be larger than another, since it is not a number to begin with. If we hold that the Lords power is infinite, certainly it would not be a countable one, but true infinity.

Lastly, since true infinity has no end, to creating something of infinite weight would mean you would have to create a universe or place that can hold it with a “larger” infinity, which is not possible. Thus, if infinity is not a number we cannot say, something is larger than infinity; it is like asking what is longer than an eternity. Perhaps it comes down to the state of being, what is the being of infinity? Perhaps we can ask, can the Lord duplicate his essence, and with safety, we can say yes. We can talk of more than one thing in mathematics that is truly infinite and employ that same infinity in various calculations; but we cannot change the nature of infinity itself, if that makes sense.

Ps. For some reason there is no notification when new posts arrive on this thread, can someone help, please?
We are a frozen spirit; our thoughts a cloud of droplets; different oceans and ages brood inside – where spirit sublimates. To some our words, an acid rain, to some it is too pure, to some infectious, to some a cure.
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Felix
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Re: Questions about omnipotence paradox

Post by Felix »

ChanceIsChange: This is something that I think should be realized much more widely (i.e., that "Omnipotent Being" is a logical contradiction). In fact, I find it inaccurate to describe God as the Supreme Being.

However, I do not see how that directly relates to the omnipotence paradox. The paradox does not make any mention of an omnipotent being. It only talks about God and implicitly assumes him to be omnipotent, but it does not claim that he is a being.
He is a discrete entity, is He not? If not, wouldn't the Omnipotence Paradox be equivalent to the problem of an Irresistible Force meeting an Unmovable Object? Is it the same problem?
Transfinite numbers, for example, are not finite but still abide by the laws of logic.
I don' think we can assert that they "abide by the laws of logic." They could be called translogical, for they are members of a set that contains an infinite number of elements, which I suppose is also true of our omnipotent God. At which point we run into the contradiction posed by Georg Cantor's "Continuum Hypothesis."

The Continuum Hypothesis

It's natural to ask if there is a set that's larger than the set of natural numbers, and smaller than the set of real numbers. The continuum hypothesis states that such is not the case. Whether this is true or false is not known, but it's unknown in a more subtle sense than that we just can't figure it out!

It turns out that a naive application of the concept of sets leads to contradictions. The simplest example goes as follows: Consider the set of all sets. Since it's a set it contains itself as an element. So it makes sense to define a set A which is the set of all sets that do not contain itself as an element. Now, if A is an element of A then, by the definition, A is not an element of A. On the other hand, if A is not an element of A, then, by the definition, A is an element of A. In either case, we have a contradiction.

ChanceIsChange: Consequently, ∞ ≤ ∞ and ∞ > ∞ may be true at the same time. It follows that (strength of God) = ∞ ≤ ∞ = (weight of stone) (so God cannot lift the stone) and (strength of God) = ∞ > ∞ = (weight of stone) (so God can lift the stone) can be true simultaneously, possibly resolving the omnipotence paradox.
Yes, this appears to be as close as we may come to a solution.... When one transcends logic, something can be simultaneously True and Not True, which, let's face it, is not a solution at all.
"We do not see things as they are; we see things as we are." - Anaïs Nin
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Whitedragon wrote:Stones in the spirit realm do not make sense, because they would have to be ethereal. It is like asking can something be a gas and a solid at the same time, or the famous question; “can the Lord create a square circle?”
I agree with you that stones in the spirit realm do not make sense. However, an omnipotent deity could still create ethereal stones that are in the spirit world. Otherwise, it would not be omnipotent. In fact, omnipotence contains the ability to make possible the impossible and even the nonsensical. Therefore, God is capable of making a square circle, since he would be restricted by the laws of geometry and thereby not omnipotent if he couldn't.

Whitedragon wrote:To get back to infinities, some infinities are larger than others, but then they are not actual infinities.
Just to clarify terminology: when you say "actual infinity", you mean "true infinity", right? I'm asking because "actual infinity" is usually contrasted with "potential infinity" and taken to mean an actual, completed infinite totality (see Wikipedia). This includes transfinite sets, which are not truly infinite according to your (and my, see nest paragraph) understanding of true infinity.

Now back to the subject. I also agree with you that those infinities which are larger than one another (e.g. the transfinite numbers) are not truly infinite because, although they have different mathematical properties from natural and real numbers, they are definite entities that obey definite laws and can be subjected to logical analysis. Georg Cantor, the discoverer of transfinite numbers, would probably agree, too.

Whitedragon wrote:True infinity cannot be larger than another, since it is not a number to begin with.
That is an interesting and important realization which I agree with, too. We should add that true infinity is also not equal to itself or unequal to itself or in any other way comparable with itself, for that matter, because it is not a number and not a definite entity altogether and self-comparability is a trait of definite objects like numbers.

Whitedragon wrote:If we hold that the Lords power is infinite
That is included in the "definition" of omnipotence and the implicit premise of the omnipotence paradox that God is omnipotent.

Whitedragon wrote:certainly it would not be a countable one, but true infinity.
Of course it would be truly infinite. (But note that there are uncountable cardinals, which are not truly infinite nonetheless.)


I still think that
ChanceIsChange wrote:(strength of God) = ∞ ≤ ∞ = (weight of stone) (so God cannot lift the stone) and (strength of God) = ∞ > ∞ = (weight of stone) (so God can lift the stone) can be true simultaneously
is essentially a paraphrasing of
Whitedragon wrote:The Lord can create a stone that is so big that he cannot lift it, but he can always lift it
expressed and clarified with more symbols. For me, the rest of your first post on this topic, including your statement that infinity could be unequal to infinity, also seems to support my opinion that I had previously paraphrased your solution and added Felix' realization that logic only applies to definite terms to justify it. Am I correct, or had I misunderstood you?


Now, however, you seem to have changed your opinion, and I prefer your new opinion to your old one as well. Originally, we essentially tried to resolve the omnipotence paradox by exploiting some properties of a semi-infinity (e.g. being greater and less than itself at the same time). Now, however, you argue that true infinity is actually beyond being greater than itself, and we have extended this to true infinity's transcendence of self-comparability above. Applied to the concrete paradox, I would say that means that, because both God's power and the stone's weight are truly infinite, it is not meaningful to say that one is greater than the other and that it is therefore also not meaningful to say that God can lift the stone or that he cannot lift it, finally freeing us from (almost) all remnants of logical constraints (hopefully).

Whitedragon wrote:would mean you would have to create
Isn't it true that God is free from all necessities because he is omnipotent?

Whitedragon wrote:which is not possible
Isn't it true that God can make the impossible possible because he is omnipotent?

Whitedragon wrote:if infinity is not a number we cannot say, something is larger than infinity; it is like asking what is longer than an eternity.
It would appear to be as you say. On the other hand, that would make infinity seem to be some pinnacle, a maximum, a typical phenomenon we encounter in the world of the strictly finite. For example, the finite set {1, 2, 3, 4} has a maximum. The countably infinite set of all natural numbers already contains no maximum, and it is a far way from being truly infinite. I prefer the following stance: it is false to say that there is an element of the set {1, 2, 3, 4} which is greater than 4, but it is meaningless to say that something is greater than true infinity.

Whitedragon wrote:Lastly, since true infinity has no end, to creating something of infinite weight would mean you would have to create a universe or place that can hold it with a “larger” infinity, which is not possible.
Concerning the creation of something with truly infinite weight, it appears to me that saying that such an object has to be included in an even larger universe is applying logic to the truly infinite, which is not valid as we saw in previous posts.

Whitedragon wrote:Perhaps it comes down to the state of being, what is the being of infinity? Perhaps we can ask, can the Lord duplicate his essence, and with safety, we can say yes. We can talk of more than one thing in mathematics that is truly infinite and employ that same infinity in various calculations; but we cannot change the nature of infinity itself, if that makes sense.
If true infinity has being (I personally agree with Felix that the truly infinite does not be, but that opinion is not rigid), then the essential question may indeed concern the nature of the being of true infinity. However, I do not understand what exactly you intend to say. Could you please clarify what you mean?
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Felix wrote:He is a discrete entity, is He not?
Does the truly infinite have attributes such as discreteness? Did any premise of the omnipotence paradox explicitly state or implicitly assume that God is an entity?

Felix wrote:If not,
How does the previous statement imply or otherwise relate to the next one?

Felix wrote:wouldn't the Omnipotence Paradox be equivalent to the problem of an Irresistible Force meeting an Unmovable Object? Is it the same problem?
I think it is exactly that. An immovable object cannot coexist with an irresistible force because a meeting of the two would cause a contradiction. Now, the omnipotence paradox arises from the fact that omnipotence implies the potential simultaneous existence of an immovable object and an irresistible force.

Felix wrote:I don' think we can assert that they "abide by the laws of logic."
The transfinite numbers are mathematical objects definable and analyzable with mathematical logic just as natural numbers, complex numbers, quaternions, matrices, sets, functions, relations etc. Can you give one example of a logical law that does not apply to the transfinite cardinals?

Felix wrote:They could be called translogical, for they are members of a set that contains an infinite number of elements, which I suppose is also true of our omnipotent God.
Could you please clarify your reasoning and what you mean?

Felix wrote:At which point we run into the contradiction posed by Georg Cantor's "Continuum Hypothesis."

The Continuum Hypothesis

It's natural to ask if there is a set that's larger than the set of natural numbers, and smaller than the set of real numbers. The continuum hypothesis states that such is not the case. Whether this is true or false is not known, but it's unknown in a more subtle sense than that we just can't figure it out!
In what way is the Continuum Hypothesis (CH) contradictory? You correctly stated the hypothesis itself and that it has been proven that it is impossible to prove or to disprove CH (in ZFC). But how is that contradictory, and how does it relate to the following Russell's paradox?

Felix wrote:It turns out that a naive application of the concept of sets leads to contradictions. The simplest example goes as follows: Consider the set of all sets. Since it's a set it contains itself as an element. So it makes sense to define a set A which is the set of all sets that do not contain itself as an element. Now, if A is an element of A then, by the definition, A is not an element of A. On the other hand, if A is not an element of A, then, by the definition, A is an element of A. In either case, we have a contradiction.
That is Russell's paradox, and it proves naïve set theory to be inconsistent. What exactly does it have to do with the omnipotence paradox other that also being paradoxical? One answer would be that just as Russell's paradox shows the set of all sets to be non-existent, the omnipotence paradox shows omnipotence to be non-existent. I outlined the benefits of this solution in one of my early posts on this topic, but I also said there why I do not like it very much. Was that what you intended to convey by bringing up Russell's paradox? I thought you had a different opinion, one that does not reject omnipotence.

Felix wrote:Yes, this appears to be as close as we may come to a solution.... When one transcends logic, something can be simultaneously True and Not True, which, let's face it, is not a solution at all.
Possibly (I still think it to be a somewhat satisfying) , but we may have found a different solution presented in the previous post based on Whitedragon's new approach. However, it may be even more fantastical than the above solution candidate and thereby even less of a solution.
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Felix
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Re: Questions about omnipotence paradox

Post by Felix »

ChanceIsChange: How does the previous statement (about God being a discrete entity) imply or otherwise relate to the next one?
Well, if God is not a discrete something (being or whatever), than what is He, an impersonal force? (like the immovable force).
ChanceIsChange: That is Russell's paradox, and it proves naïve set theory to be inconsistent. What exactly does it have to do with the omnipotence paradox other that also being paradoxical?
Just that defining the Omnipotent God as "the Set of all sets that does not contain Itself as an element" produces the same contradiction.
ChanceIsChange: Concerning the creation of something with truly infinite weight, it appears to me that saying that such an object has to be included in an even larger universe is applying logic to the truly infinite, which is not valid as we saw in previous posts.
But this is the flaw in the paradox problem to which I referred: it requires that relative finite concepts such as "weight" and "lift" be given infinite proportions, which even omnipotence cannot accomplish - the finite cannot be made infinite, it is one or the other and ne'er the twain shall meet.
"We do not see things as they are; we see things as we are." - Anaïs Nin
Socrates Tea
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Re: Questions about omnipotence paradox

Post by Socrates Tea »

Just think of it this way:

Maybe God is actually speaking reality into existence, so all he has to do is say, "A stone so heavy I can't lift it."
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LuckyR
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Re: Questions about omnipotence paradox

Post by LuckyR »

Socrates Tea wrote:Just think of it this way:

Maybe God is actually speaking reality into existence, so all he has to do is say, "A stone so heavy I can't lift it."
Or maybe, god claiming (or his minions in his stead) omnipotence, is lazy shorthand for: "I am way, way more powerful than you, so much so that it seems like omnipotence from your perspective, though I am not actually omnipotent".
"As usual... it depends."
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Felix wrote:
ChanceIsChange wrote:How does the previous statement (about God being a discrete entity) imply or otherwise relate to the next one?
Well, if God is not a discrete something (being or whatever), than what is He, an impersonal force? (like the immovable force).
Now I understand the connection you mean. Well, I would say that God simply is God.

Felix wrote:Just that defining the Omnipotent God as "the Set of all sets that does not contain Itself as an element" produces the same contradiction.
So it would seem. However, sets are entities subject to logic, whereas I argue that omnipotence isn’t.

Felix wrote:But this is the flaw in the paradox problem to which I referred: it requires that relative finite concepts such as "weight" and "lift" be given infinite proportions, which even omnipotence cannot accomplish
Omnipotence can accomplish anything at all by “definition”.

Felix wrote:the finite cannot be made infinite, it is one or the other and ne'er the twain shall meet.
Firstly, omnipotence can make the finite infinite by “definition” (because it can achieve anything by “definition”), and secondly, wouldn’t drawing a line between the finite and the infinite actually be a limitation of the infinite, which would, of course, be contradictory to the concept of true infinity?
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

Socrates Tea wrote:Just think of it this way:

Maybe God is actually speaking reality into existence, so all he has to do is say, "A stone so heavy I can't lift it."
Yes, and then that stone would instantly come into existence, whereupon God would realize that he can’t lift it, which runs contrary to his omnipotence. That is exactly the paradox we are trying to resolve.
ChanceIsChange
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Re: Questions about omnipotence paradox

Post by ChanceIsChange »

LuckyR wrote:Or maybe, god claiming (or his minions in his stead) omnipotence, is lazy shorthand for: "I am way, way more powerful than you, so much so that it seems like omnipotence from your perspective, though I am not actually omnipotent".
That is not what the omnipotence paradox is about. The omnipotence paradox is used to investigate the nature of omnipotence, and ‘God’ is just shorthand for ‘omnipotent “entity”’.
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If Life Stinks, Get Your Head Outta Your Buts
by Mark L. Wdowiak
September 2021

The Preppers Medical Handbook

The Preppers Medical Handbook
by Dr. William W Forgey M.D.
October 2021

Natural Relief for Anxiety and Stress: A Practical Guide

Natural Relief for Anxiety and Stress
by Dr. Gustavo Kinrys, MD
November 2021

Dream For Peace: An Ambassador Memoir

Dream For Peace
by Dr. Ghoulem Berrah
December 2021