Can God defy logic?
I was wondering whether there is an argument in favor or against this statement: "God can defy logic".
This issue raised up in another thread. I was questioning whether 1+1=3 is possible.
This issue raised up in another thread. I was questioning whether 1+1=3 is possible.
Comments (118)
Quoting JustSomeGuy
When you ask about the things God can do, you have to first make it clear what characteristics your God has.
If God is omnipotent, then yes, he can defy logic.
This is a symbolic convention. It is not logic.
As a an aside, there are no two things that are exactly alike so we are doing arithmetic everything out of convention.
Quoting bahman
I don't think even an omnipotent being could defy hard rules of logic. 1+1=3 is not possible in the traditional system of mathematics and it would not be any more possible for us than for God.
However, I don't think any legitimate definition of omnipotence would require a deity to be able to defy th laws of logic. I don't even think that 1+1=3 is even intelligible. What would it mean? What would it look like? What would it represent?
1.) Everybody agrees on what one of something, S, looks like.
2.) Everybody agrees on what three of S looks like.
3.) You combine one of S with another one of S and get three of S.
If you don't know what I mean, imagine somebody pouring one liter of water into a container with one liter of water and it resulting in three liters of water. That does not happen in our world, but that does not mean that there are not other worlds where it happens.
Quoting bahman
If yes, then what is the point of the free will?
If no, then what is the point of the free will?
Could God create a rock which was so heavy that even he could not lift it? If he can't then he is not omnipotent because there is a logically possible object that he cannot create, if he can, then he can't be omnipotent because there is a logically possible task he cannot do.
The counter argument to this is that such a rock is not logically possible because a rock which is so heavy it cannot be lifted by an omnipotent being could not exist, it would have to be more than infinitely heavy.
So the problem arises, if God is not bound by logic, then where does that leave the refutation of the limits to his omnipotence. If he is not bound by logic, then he could create such a rock which would instantly disprove his omnipotence.
So your question amounts to "Can god do things that defy description?"
And that's a dead question - unless you care to set out a description of something that defies description...
Another thing is even our thoughts aren't "constrained" by logic. We all have some inconsistent beliefs floating around somewhere in our minds.
I don't see why God, an omnipotent being, should be limited by logic.
I am arguing that 1+1=2 is symbolic equation which tells something we can assign to reality we observe, one apple plus one apple is equal two apples. The question is can we have 1+1=3 which again can appear as a property of reality? Is that possible?
Quoting Rich
I don't understand you?
I agree.
Quoting Brian
I agree.
Sort of. There are no two things that are equal, so the equation doesn't represent anything we observe. It's a practical symbolic representation of an idea. Immobile symbology can never capture the motion of the universe, it can only approximate. Unfortunately, academic philosophy (for the most part) teaches the symbol is what it's real. It is actually just a convenience for communication.
One can say 1+1=3 as a cipher if one wishes, the meaning being deciphered by the receiver. It is just communication symbols, that is all.
I don't think such a world exist.
What free will has to do with our current discussion?
I cannot agree more.
Think about it (Only if you want).
1+1=2 represent something in reality.
To me such a person is impossible.
If movement is possible then there's a problem with Zeno's paradox. The issue we have is in figuring out what that problem is.
There is no such a thing as Zeno's paradox. You can move along infinite instants if the duration between instants is good function of distance.
Quoting TheMadFool
Self-contradictory ideas cannot exist.
Quoting TheMadFool
Because such an illogical reality is impossible.
I believe he was referring to cognitive dissonance, which is a real phenomenon.
I am sorry. I have to correct myself. I wanted to write "Self-contradictory ideas cannot exist in reality". They are impossible in any world. Cognitive dissonance is a mental phenomena.
If God cannot defy logic, then God is not omnipotent. Plain and simple. A God who can defy logic would be more powerful than a God who cannot, and omnipotence means "all-powerful".
If you're going to claim God cannot defy logic, you're claiming God is bound by the laws of the universe he himself created (assuming we're talking about a creator God).
With all of this...
...you're talking about things that are not possible in our universe. These things are not possible because of the way our universe operates. We cannot say anything about what any other possible universe/reality could look like, because we only have experience of our own. We are limited by our universe. God is not.
If you believe God is limited by our universe, then he cannot be omnipotent, and he cannot be the creator of it. That's fine, you can believe in that sort of God, but when people speak about God they're usually speaking about an omnipotent creator God.
Doesn't follow. He might have had the power to create it but not have the power to do whatever he likes with the result.
If defying logic is nonsense then to say that a God who can defy logic is more powerful than a God who can't is like saying that a four-sided triangle has more sides than a three-sided triangle.
The triangle with the most sides has three sides, because a triangle with more sides is nonsense. The God with the most power has the power to do anything that is logically consistent, because a God doing the logically inconsistent is nonsense.
If you're just going to define "omnipotence" in such a way that it requires the power to defy logic then the problem is with your definition, not with the notion of a creator God.
I'd like to see an argument for that as a reasonable possibility. If God created the universe and all of the laws by which it operates, what reason would there be to believe that he could not control it once it was created? Seems baseless, to me.
Quoting Michael
It's not at all like that. Claiming that a God who can do something that another God cannot is more powerful, would be equivalent to saying that a shape with more sides than another shape has more sides. It's essentially a tautology.
Claiming that "defying logic is nonsense" is also a tautology. You're just describing what "defying logic" means. The claim that "defying logic is nonsense" isn't saying anything at all in regards to the issue at hand.
Does God make judgements? If so, then does it not use logic to do so? Does God create, or have a causal effect on the things around it? If so, then there is a logical process to the acts of God.
So it doesn't seem like a valid question to ask if God can defy logic, as God defines what is logical. There could never be such thing as God being illogical, or defying logic. God would be the source of what is logical, always.
Interesting perspective. Very similar to the morality issue.
There is no God who can do the logically impossible, as such a thing is incoherent. There is just (possibly) a God who can do the logically possible, and that's as powerful as anything can possibly be.
You're begging the question. You're saying that a God that defies logic isn't possible because it would defy logic.
Okay, you don't seem to be understanding what I'm saying, or what you're saying yourself.
Let me ask: what do you mean by "incoherent"? Define the term.
In other words, you think that the laws of physics that you have been taught apply to all possible places in all possible times.
How do you know?
The laws of physics could be different but they should respect something. "One apple+one apple=two apples". I cannot even imagine "one apple+one apple=three apples".
So things you can't imagine can't be possible for God? God is bound by your imagination?
I am saying that something which is unimaginable cannot have any instance in reality. And yes, if you give two apples to God He only have two apples.
This is the root argument everyone who says "God cannot defy logic" is making here.
You're saying that God is limited by what you can imagine. Why on Earth would that be the case? It's quite egotistical, isn't it? I'm not calling any of you egotistical, just so we're clear; I don't believe any of you are making the argument with the intent of it being egotistical. But that doesn't change the fact that it is an egotistical claim.
This may be a cliche kind of metaphor, but consider something "lower" than us. Lower than biological life forms. A rock, for example. Do you think a rock understands logic? Can a rock imagine mathematics? If course not. It's a rock. We are not rocks, which is why we can do those things that rocks cannot conceive of (including conceiving of things).
Now, when we talk about God we're talking about a totally different "kind" of thing than we are. Similar to rocks vs. humans, but to an even greater degree. Why in the world would this thing (God) be constrained by our limitations?
Well, I can imagine combining a cup of water with another cup of water and getting three cups of water. Therefore, it would be 1 + 1 = 3.
We tend to be quite selective in choosing our examples. 1+1=1 when we watch raindrops slide down a window. 1+1=3 when folk fuck without contraception.
What you are doing is merely redefining the concept of "1" so your theory applies. Two drops of rain of course could form one but the quantities of water satisfy the equation 1+1=2. So I think we need to be clear about what we mean with concepts used in left and right hand side of equation. All I am saying is that 1+1=/=1 if the "1" in left side of equation is equal to the "1" in right side of equation.
Exactly.
And that is exactly what we have done; defined "1" so that the theory of numbers applies.
If god 'defied' logic, we could simply change the rules of logic so that they applied to his actions.
That is, the question in the title badly misunderstands what logic is.
To put it another way, logic does not constrain the world in any way, nor does it constrain God. It constrains what can be sensibly said. It declares, for instance that, it does not make sense in the context of omnipotent beings, to talk about stuff they cannot do. So don't do it.
Is 1+1=1 if 1 in both side are same?
That's a pretty ambiguous question. Both sides of what - the '+' or the '='? I gave you an example where the equation worked with drops of water. One drop of water added to another drop of water makes one drop of water.
If a square is a regular polygon with four equal sides then I don't know of any way to make it a triangle.
But if it is a regular polygon in which all four angles are 90º?
Drawn on a sphere, it would have three sides.
And if we insist on both 90º angles and four sides, we constrain the space to a plane.
Oddly, logic constrains the world...
Banno answered this pretty well, I think, but I'll refer you to another post I made which covers this also:
Quoting JustSomeGuy
To elaborate for your specific question, in our universe a shape cannot have four sides with all right angles and three sides with non-right angles at the same time. The fact that this is not possible in the universe we live in says absolutely nothing about what God is capable of, because God is not bound by our universe if he is omnipotent and created it. Just because something doesn't make sense to our very limited human brains, doesn't mean it is beyond the capabilities of God. I already discussed this in another post, as well (I suggest you actually read through the thread next time):
Quoting JustSomeGuy
That's not something I would agree with.
Just to sure I am not misunderstood, a square triangle cannot exist on a plane; and that's jut a result of what a square and a triangle are. Further this is so in any possible universe.
Well, in a sense I agree and in a sense I don't. It really has to do with language. The language we are using describes concepts in this universe; we cannot apply any of them to any other possible universe or anything outside of our universe. Like I said earlier, we cannot say anything about anything other than our own universe, because our universe is all we know. Our language and our logic are products of our observations of this universe. It's exactly as unenlightened said:
Quoting unenlightened
Of course we can. Hence modal logic.
I suspect @unenlightened might well agree with me if I pointed out that a world containing square triangles would not be talking about the same squares and triangles that exist in our world. The meanings of these words would have changed, as the meaning of triangle is changed when a triangle is scribed on a sphere.
Logic does not constrain the world in any way. But sometimes the logic one chooses is the wrong one for what one has to say, and hence needs changing.
That's exactly what I was trying to say; maybe I should have elaborated more.
Quoting Banno
This follows from what I said, that logic is a product of our observations of this universe.
I think we agree and I just wasn't clear enough.
If by 'God' you mean a perfect being, then no I don't think so. If god is perfect he can have no flaw and not being logical is a flaw. If perfection is the essence of god's being then god cannot defy logic, because he can't defy his essence, which is always in perfect balance.
1+1=3, I heard a joke like that once. A business man was interviewing for a new accountant. He asked each person he interviewed 'what does 2+ 2 =?' and they all said 4 , then late in the afternoon when he asked the question to one of his prospects , the prospect went over to the window and pulled the blind down, turned and asked the business man "what do you want 2 + 2 to equal?" He got the job.
"Perfect" and "flaw" are both subjective concepts.
(¬(G = G)) ? (G = G)
Not sure why anyone would make such a move; they'd no longer be talking about anything in particular.
Weird :o
All concepts are subjective.
What? Does that mean?
So does it make sense to apply our subjective concepts to God?
Perfect in regards to what? As I said (and you agreed), "perfect" is subjective. So when you say something is perfect, you need to make explicit what it is perfect for.
Again, as I said (and you agreed) "flaws" or "defects" are also subjective. What is all of this in reference to? Perfection and flaws only make sense in context or comparison.
I'll give you an example:
A perfect sphere is only perfect in regards to it matching every single one of the criteria for what a sphere is. A sphere would be flawed or have defects if it met some, but not all, of these criteria.
So, when you say God is "perfect", what is he a perfect example of? Furthermore, what criteria is God meeting that makes him perfect? You cannot have perfection without initial conditions being set, because perfection as a concept is relational to specific criteria.
Zeno's paradoxes
Quoting bahman
Theism vs Atheism...Materialism vs Idealism...etc.
Quoting bahman
Yes, the current state of logic doesn't allow contradictions but people have developed different kinds of logic that allow contradictions - see paraconsistent logic.
Perfection is only understood in regard to what is not perfect.
I just wanted to show the OP that contradictions aren't impossible.
Quoting TheMadFool
"practice makes perfect"
Perfect for what????
The question makes no sense.
Perfection is a state of being, it is not for anything except itself.
Are you serious?
Okay, I give up. I tried.
Well tell me what do you think god is?
"So close yet so far"
That is the perfect song for this occasion (though I find Elvis to be perfect for any occasion)
Must be a coincidence. :)
Nature.
I thought of one more way to try to get this concept across:
Remember those spheres I was referring to earlier? I said one was a perfect sphere and explained why. But you say God is perfect, so I assume this hypothetical sphere can't be perfect, because if it were it would be God according to you, correct? So tell me, what is it that makes my sphere imperfect? What characteristics does it lack which are required to qualify for your idea of perfection?
I guess
Actually I don't disagree, I have recently thought that some sort of plural pantheism might be close to the case.
However, I took this discussion to be more of an epistemological discussion about the nature of god, as the absolute, as a subjective concept. Your sphere participates in god's perfection, as Plato would have it.
Read Spinoza's Ethics, if you haven't. His explanation of God as Nature had a huge impact on me.
Quoting Cavacava
It was. I was just trying to get you to understand that perfection needs qualifiers. You cannot just say something is perfect, you have to be clear on what it is a perfect version or example of, and what characteristics qualify it to be a perfect version or example of whatever it is.
You should look into this more elsewhere, maybe Stanford Encyclopedia of Philosophy. The things I'm saying are true; this wasn't an opinion I was defending. You just don't seem to understand the concept of perfection.
Reading William James.
I guess we will have to disagree here, I don't think that perfection can be qualified or limited in any manner.
ps. good night.
I don't understand the distinction you're trying to make here with regards to the use of logic to resolve the paradox. If God can create such a stone, the it becomes logically possible for such a stone to exist (by God having created it). He doesn't have to actually create it for the logical possibility to derive. So as soon as we admit that God can create such a rock, the exists a logically possible thing that he cannot do. Whether he actually does it not is irrelevant.
But this doesn't tell us anything about God, it tells us about the uselessness of the term 'omnipotence'. It's like 'infinity', it's a perfectly logical term, but when you start to actually play with it, the meaning starts to break.
We could do the same stone paradox in any number of ways to eliminate the 'can/actually does' problem. Can God create a being more powerful than himself? If he can't then he is not omnipotent because we can conceive of a task he cannot do, if he can then he's not omnipotent because we (or he) can conceive of a being with greater powers.
Can God create a universe in which his existence is unnecessary? (how do we know he hasn't already done so?). Can God reverse time and create a universe such that, on going forward in time he would cease to exist? Can God create something capable of destroying God? I mean we could go on all night. Omnipotence is a meaningless concept unless constrained by what is logically possible.
If we're to say that God can defy logic, but then admit we are incapable of conceiving of such an entity, then we are not really making a statement about 'God' (the concept in our heads) at all, because that concept is constrained by the ability of the human mind to conceive of it.
God then, if we pursue this unconstrained approach, ceases to be a meaningful concept and just becomes a place-holder for "all the things we can't conceive of", which is certainly a far cry from any religious interpretation of the deity, even Spinoza's God ceases to have any definition if it no longer is constrained by conceivability.
At T[sub]1[/sub], before creating the stone, there's nothing he can't do. At T[sub]2[/sub], after creating the stone, there's something he can't do. Therefore, at T[sub]1[/sub] he's (accidentally) omnipotent and at T[sub]2[/sub] he's not omnipotent.
You seem to be saying that if it's possible that at T[sub]2[/sub] there's something that God can't do then God isn't omnipotent at T[sub]1[/sub]. That would be essential omnipotence, which personally I think is a useless definition. If at T[sub]1[/sub] there's nothing God can't do then why not just call him omnipotent at T[sub]1[/sub]? That he could (not even will) limit his own power in the future doesn't take away from his power in the present.
You are telling me that our insistence obliges the world to be of some geometry and not another? I think not. No, our insistence constrains what we can say about the world, and possibly makes it impossible to tell it like it is. The world shrugs and carries on as it pleases.
Consider all "1" similar.
But then I'm omnipotent right now. There's nothing I can't do right now until some of the Atoms in the universe organise themselves in such a way as to prevent me, which of course they will do the instant I try any impossible action, but right now I can do anything. Humes's problem of induction.
This seems facetious. If God wants to create or destroy or rearrange a world in any conceivable way then he can. He can do anything. That he could choose to limit his power doesn't then entail that his power is currently limited. He's all-powerful. Any other definition of "omnipotent" seems pretty pointless.
And on the same point, to say that a God that can't limit his own power (by creating a stone too heavy for him to lift) isn't omnipotent is also to use a pretty pointless definition of "omnipotent".
It's up to you if you want to use a definition of "omnipotent" that is self-defeating or open to paradox, but why would you want to do that?
This is begging the question. That's what we're trying to find our, if it makes any conceivable sense that God is both omnipotent and unconstrained by logic. If you're starting from the premise that God definatly is omnipotent then you've already ruled out one of the options.
Quoting Michael
He is not 'choosing' to limit his power by creating the stone, it is limited already by the logical possibility of creating such a stone.
You'll have to explain why you think such a definition of omnipotent is 'pointless' as opposed to yours because I'm not seeing any reason.
Sorry if I wasn't clear. What I'm trying to say is that if God can will a universe into and out of existence, and if he can will it to behave however he wants, then it is entirely appropriate to consider God to be omnipotent, even if it is (im)possible for God to create a situation in which there is something he can't will to happen (e.g. lift a particular stone).
Because as the paradox tries to show, such a definition is self-defeating. Your definition of "omnipotence" depends on a contradiction: it is both the case that God can create a stone that he cannot lift and the case that it isn't possible for there to be a stone that God cannot lift (reminds me of this comic). Why choose to use an incoherent definition?
Like above, your definition of "unlimited" power depends on a contradiction, so it seems like a pointless definition. I think it far more reasonable to say that to have unlimited power is to have the power to will a universe into and out of existence, to have the power to will the universe to behave however one wants, and for there to be no other power to stop you. The mere (im)possibility of being able to create a situation in which one cannot do something isn't sufficient to deny this unlimited power.
You've pretty much nailed what I am trying to say (although I personally have jumped the other side of the fence on the use of the word omnipotent in that I think it is is more useful philosophically to point out it's incoherence than to define it in a useful way). Putting all that aside, your cartoon is as good a summary of what I'm saying as any.
Of course God had to be able to defy logic, because if he wasn't able to, his omnipotence would be constrained by the rock problem. His only way out of it is to say that he can create a rock he can't lift and then lift it, because he's not constrained by the laws of logic that say he can't, or that he can create such a rock because he's not constrained by the fact that such a rock is a logical impossibility. Either way he cannot be both omnipotent and constrained by logic.
Or we might say that if God defies logic, he is not God. Because logic ensures that God is God.
I am aware of that. But the paradox is resolvable considering the fact that we need shorter duration for shorter distance.
Quoting TheMadFool
I should have written Self-contrary ideas cannot exist in reality.
SO He can and cannot lift the stone at the same time.
I think you're both missing the point, though. If we could make sense of God (the type of God we're discussing), then it would not be God. A necessary characteristic of a God which exists apart from the universe is that we cannot make sense of it. If we could, it would necessarily be part of our universe, because as I already said earlier: we cannot say anything about anything apart from our universe and our experience of it. So, if a characteristic of God is that it is not part of the universe, then it follows necessarily that we cannot say anything about what this God is like.
If we can talk about God as being transcendent and the creator of the universe then why can't we talk about God as being all-powerful (whatever we mean by the term)?
Quoting JustSomeGuy
Here:
If God created the universe, then God is not part of the universe
We cannot say anything about anything apart from our universe
Therefore, if God created the universe, we cannot say anything about God
This is a logically valid argument. You can argue either of the two premises, but it logically follows that IF God is separate from the universe AND we cannot say anything about anything apart from our universe, THEN we cannot say anything about God.
If we cannot say anything about God then we cannot say that God created the universe. Your argument refutes itself.
Well, in a sense, yes. That is essentially what I mean. If we have no experience of something, we cannot say anything about it. Meaning, nothing anybody says about it can be said to actually apply to it. So, if a condition of your God is that it created the universe, you cannot say anything else about that God.
Quoting Michael
I understand your reasoning, but that doesn't disprove the argument. If the argument is sound, and your conclusion is correct, it's still showing what I was intending to show: that discussing a creator God is impossible/illogical. That's not to say a creator God cannot exist, only that if one does we cannot talk about it. As I've said before, this all really comes back to the limitations of language as well as the limitations of our brains.
If it helps to replace "say" with "know" in my argument, that's fine. Maybe that's where the confusion lies. I just don't like to use the term "know" because that just opens up a can of worms about what knowledge is.
But since you seem to take issue with my use of the term "say", then go ahead and replace it with "know".
Having abandoned logic, you've abandoned identity ("it's fine saying that G is not G"), non-contradiction (G is possible and impossible, G both refers and does not), and some other things.
So, with G there's no longer anything in particular that's talked about.
Maybe G is just goat. (And not goat, but I prefer goat.) :)
To me it seems mostly like linguistic suicide, baby-talk, shooting oneself in the foot.
Which sort of makes me wonder what people are talking about here, well mostly the Yes voters.
I agree with your exposition, the logic is infallible, for your definition of God, but I don't agree that such a definition is ever the one we're using. I think that because God is a concept in our minds he can only have properties that we can conceive.
We can talk about a God that has other properties, in the same way as we can say "this statement is false", but we are not really imagining him with those properties because we can't.
We can also understand the concept that some things are outside of our understanding, but then, I think it would be disingenuous to all religion to suggest that these things are what they mean by God. God speaks to people, sends thunderbolts, works out moral systems, decides what should happen adulterers. He's very much of this world, intrinsically tied up with human affairs.
So, I agree with you entirely that a creator of the universe must logically be outside of our conception, but I don't really think a creator of the universe (complete with all the logical implications you outline) is what anyone actually has in mind when they talk about god.
Exactly. Basically all I've been trying to say is that we can't talk about a God outside of our universe, so it follows that people discussing God in these ways cannot be talking about a creator God. Really, I just wanted to show the inconsistency there that most people don't seem to realize.
Quoting Pseudonym
I'm sure it's not what they have in mind because they haven't thought through the logical implications. I was just trying to bring some of those implications to light and show that talking about this kind of God at all is incoherent. We can really only talk about a God that is part of our universe, but this means that God could not have created the universe. Most people who believe in a God hold these incompatible beliefs about God.
I guess the conclusion to all of this is that if you're going to believe in a creator God, you can't talk about it at all because doing so limits it to our universe.
Edit: I just realized I'm basically describing the first few lines of the Tao Te Ching.
"The Tao that can be spoken is not the eternal Tao
The name that can be named is not the eternal name
The nameless is the origin of Heaven and Earth"
I agree entirely but it raises an interesting question of whether any one could actually believe in a creator God at all. Sure, they could 'say' the words, but if we've just agreed that a creator God is not something of which the human brain could conceive, then how could anyone believe in it? Can one really believe in something one cannot even conceive of? I'm not sure that sounds possible.
That is an interesting question, indeed. I think it's probably possible to believe in something you don't understand. Many people believe in gravity without actually understanding it. But I suppose that's really believing in a representation of something, and not the thing itself.
You may be right; on its face it doesn't seem possible, though it would probably require more discussion of what "belief" actually is.
I think the problem is best resolved with the dispositionalist approach to belief of people like Ruth Marcus. By this approach, people do not believe in gravity, rather they believe that gravity will produce the effect of returning the ball to earth.
We could use this approach to allow that people do not believe in God at all, but they believe that God created the universe, in which case they need not conceive of the properties of such an entity, only its actions.
I agree, that approach does resolve the problem quite well.
The paradox hasn't be resolved.
Quoting bahman
Contradictions are worse than contraries.
So you didn't get my point?
Please explain how Zeno's paradox is solved.
t=x/v where x is distance, v is speed and t is duration. Does it take shorter duration if the distance is shortened and speed is constant? Yes. Therefore you spend less time while you divide a distance into two and that allows you to move from the beginning to end during a specific time duration. That is true that number of interval is infinite but the sum of infinite interval is finite.
Apologies if this has been covered already, but which paradox are you referring to? There are many.
You are right. There are many paradoxes. But I think we are discussing about the paradox related to the runner.
It's not about infinite steps. You can't make the first step. To move a distance of 1 meter you must first move .5 meters. To move .5 meters you have to first move .25 meters and so on ad infinitum. Can you tell me the exact size of the first step you'll make to travel 1 meter? You can't because distance can be divided infinitely.
Quoting JustSomeGuy
Please read above.
Movement is not discrete but continuous.
If we're talking about Achilles and the Tortoise, the solution is simple. An infinite number of points can still add up to a finite distance. Just as you can divide a square in to an infinite number of smaller squares, and yet the sum of the area of all of them will still be the area of the original square, which is finite.