Introduction to Lesson:

Today we are going to talk about paradoxes, and we are going to discuss some examples together.  

First, what is a paradox? Has anyone heard of paradoxes before?

[Invite student responses].

A paradox is a special type of puzzle or riddle. It’s when two things seem to be true at the same time, but it’s really hard to understand how they can both be true, because they seem to contradict, or go against, one another.

To understand paradoxes better, we are going to talk through some examples.

Superhero/Omnipotence Paradox

We are going to start with the Superhero Paradox (it is sometimes called the omnipotence paradox or God paradox, because it’s about someone who is very very powerful – omnipotent means all-powerful).

So, let’s say there is a superhero, and this superhero is so powerful that they can do anything and everything. Take a minute and imagine your superhero in your head. What do you want to ask them to do?

[Students may volunteer ideas].

Offer questions:

If I asked the superhero to create a 9-million-foot-tall balloon animal, could the superhero do that? Yes, remember, the superhero can do ANYTHING.

What if I asked the superhero to drink all of the water in all of the oceans of the world? Our superhero can do that too – no problem!

What if I asked the superhero to make a huge boulder that turns into cotton candy when someone touches it; could s/he do that?

Now – what if I asked him/her to make a boulder that is so heavy (SO HEAVY!) that s/he cannot lift it when s/he tries. Can the superhero make that boulder?

[Invite yes/no responses from students].

If yes – how is it possible that s/he can’t lift the boulder? I thought we said s/he could do everything?

If no – how come s/he can’t make a boulder that heavy? I thought we said s/he could do everything?

[Invite discussion from students.]

Depending on students’ ages/grade level, level of understanding, interest in the topic, and time available, this may lead into a discussion of the concept of omnipotence and its limits, including tasks that are logically possible vs. physically possible vs. technologically possible within a given world. For example – could our superhero make it such that 2 + 2 = 5? Why or why not?]

Sorites Paradox:

The name of this next paradox, the Sorites paradox, comes from a Greek word, soros, meaning pile or heap, and we’ll see in a minute why this paradox is so named.

Language and the words that we use can be specific, or they can be vague. When we use specific language, we can be pretty sure that we know what the speaker or writer means, and we can confidently point to something in the world to go along with the word(s) that are being used. So, if someone says, “I have a dozen bagels,” then we know that they mean that they have exactly 12 bagels. So, the word dozen cannot apply to 11 bagels, nor 13 bagels. We know exactly what “a dozen bagels” refers to. (At least in terms of the number – we don’t know what assortment of delicious flavors are contained in the dozen!) J

Vague is when a word doesn’t pick out something specific in the world, but rather we have to make a decision for ourselves about how we use the word – it’s a little unclear. Has everyone here heard of the word “a bunch”? Can someone give us an example of a sentence using “a bunch”?

If someone says they have a “bunch” of bagels, how many do they have?

If they have one bagel, is that a bunch of bagels? Why or why not?

If they have two bagels, is that a bunch of bagels? Why or why not?

What about four?

What about a million bagels? Do you think the word “bunch” still makes sense if you have a million bagels?

Do you think there is a specific number we can say is the line or the marker between one bagel and a bunch of bagels? How do we know when to use the word?

[Invite discussion/debate from students].

• How come we use the word “bunch” in our daily lives rather than using specific numbers, if “bunch” causes so much disagreement?
• Are vague words useful? Why or why not?
• What about a man with no hair on his head – what do we call that? (Students may respond with “bald”).
• Now, what if he has just one hair, just one, and you can barely see it? Imagine that man in your mind. Is he still bald? What about two hairs?
• Is there a specific number of hairs that suddenly changes the man from bald to not bald?

These questions are examples of, or variations of, the Sorites paradox, which originally used the idea of a pile of sand and the number of grains it takes to form a pile.

So let’s say you’re trying to build a big pile of sand, and I give you just one grain. Do you have a pile yet? What about if I give you a second grain, so now you have two – is that a pile?

We may feel that we know what the word pile means, and we can picture a pile in our head, and we know that at some point, if I keep giving you one extra grain of sand, and another, and another, it will turn into a pile. But what number of grains turns us from not having a pile to having a pile? If we can’t pick out a specific number, then do the grains ever really turn into a pile? When?

Why does the Sorites paradox matter? Can you think of an example of a real-world problem where we have to decide on a cutoff, but it’s kind of hard to decide where that cutoff is?

[Invite student answers].

Sometimes we may not have really good reasons for why we decide to place the cutoff at one point versus another. Could this cause problems in the real world? When or why?

[Invite student answers].

Offer example:

In the United States, we let 18-year-olds vote in elections, but we do not let people vote who have not yet reached exactly 18 years. Do you think that a person who is 18 years and one day old is better able to think about the policy issues or the candidates and make an informed choice compared to someone who is 17 years and 364 days old? Is it okay that we let the person who is just two days older vote, whereas the person who is two days younger cannot vote and has to wait until the next election to cast their ballot? Why or why not?

Now, I invite you to think of more examples of the Sorites paradox! Do our cutoffs have significance, or importance, in the real world? If so, how can we justify, or give good reasons, for where we draw our cutoffs?

Zeno’s Dichotomy Paradox

An ancient Greek philosopher named Zeno came up with four famous paradoxes. We are going to focus on just one today, called the Dichotomy Paradox. A dichotomy is when something can be divided into two main parts, or when something is usually thought of in terms of two major forces – like good and evil, or war and peace, or happy and sad. But today, we can call this paradox the “forever cake paradox.”  J If you don’t like cake, think of something else you really like to eat that can be cut nicely. Maybe a pizza!

So, let’s say that you have a delicious cake, and you love this kind of cake so much – it is your favorite flavor. And you decide that every day, you are going to cut the cake in half and eat just one half, until it’s gone. So on the first day, you have your whole cake, and you cut it into two halves, and you eat one half. So what is left? Yes, the other half is left!

The next day, you take the piece you have left over out of the fridge, and you cut it into two halves again. And you eat one of the halves, and so the other half is left in the fridge. And then the next day, you do the same thing, and you cut it into two halves, and you leave one of the halves for your next day’s snack.

Will you ever run out of cake? Why or why not? If you cut the leftover cake into two pieces every day, you can always leave the other piece. So how could you ever run out of cake?

[Invite student discussion. Depending on students’ grade level and interest, may use mathematics to explore the paradox further.]