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Title: Natural Numbers (in Set Theory)
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Series: Start Learning Numbers
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Parent Series: Start Learning Mathematics
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YouTube-Title: Start Learning Numbers 1 | Natural Numbers (in Set Theory)
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Subtitle in English
1 00:00:00,280 –> 00:00:02,049 Hello and welcome to
2 00:00:02,059 –> 00:00:03,779 start learning numbers, a
3 00:00:03,789 –> 00:00:05,039 video course where we will
4 00:00:05,050 –> 00:00:06,320 discuss all the different
5 00:00:06,329 –> 00:00:08,130 sets of numbers starting
6 00:00:08,140 –> 00:00:09,220 with the natural numbers
7 00:00:09,229 –> 00:00:10,899 and going all the way until
8 00:00:10,909 –> 00:00:12,579 we reach the complex numbers.
9 00:00:13,100 –> 00:00:14,420 But first, I want to thank
10 00:00:14,430 –> 00:00:15,659 all the nice people that
11 00:00:15,670 –> 00:00:17,180 made this course possible
12 00:00:17,190 –> 00:00:18,649 by supporting me on Steady
13 00:00:18,659 –> 00:00:19,459 and paypal.
14 00:00:20,180 –> 00:00:21,870 When you first see mathematics,
15 00:00:21,879 –> 00:00:23,819 you often see that mathematics
16 00:00:23,829 –> 00:00:25,159 starts with calculations,
17 00:00:25,170 –> 00:00:26,959 calculations with numbers.
18 00:00:27,629 –> 00:00:29,329 And the numbers start with
19 00:00:29,340 –> 00:00:30,370 the natural numbers,
20 00:00:31,219 –> 00:00:33,069 they are called natural numbers
21 00:00:33,080 –> 00:00:34,229 because they are an
22 00:00:34,240 –> 00:00:36,110 abstraction we can immediately
23 00:00:36,119 –> 00:00:37,639 see in our natural world.
24 00:00:38,430 –> 00:00:40,090 In fact, even as Children,
25 00:00:40,099 –> 00:00:41,810 we already get the conception
26 00:00:41,819 –> 00:00:43,159 for counting things
27 00:00:43,959 –> 00:00:45,250 and the abstraction or
28 00:00:45,259 –> 00:00:46,930 generalization that comes
29 00:00:46,939 –> 00:00:48,740 from counting different objects
30 00:00:48,750 –> 00:00:50,599 is what leads us to the natural
31 00:00:50,610 –> 00:00:51,220 numbers.
32 00:00:51,840 –> 00:00:53,720 So they just represent different
33 00:00:53,729 –> 00:00:55,700 sizes we can have for collections
34 00:00:55,709 –> 00:00:56,659 of some objects.
35 00:00:57,419 –> 00:00:58,810 Therefore, it makes sense
36 00:00:58,819 –> 00:01:00,419 to introduce new symbols
37 00:01:00,430 –> 00:01:02,069 for these representations.
38 00:01:02,549 –> 00:01:04,029 Now please keep in mind for
39 00:01:04,040 –> 00:01:05,680 the mathematics we will explore,
40 00:01:05,690 –> 00:01:06,860 it does not matter which
41 00:01:06,870 –> 00:01:08,580 name we give these numbers.
42 00:01:08,589 –> 00:01:09,940 But the actual concept of
43 00:01:09,949 –> 00:01:11,910 numbers is important with
44 00:01:11,919 –> 00:01:13,489 this, you could say the natural
45 00:01:13,500 –> 00:01:14,989 numbers we want to consider
46 00:01:15,000 –> 00:01:16,949 is the set N that consists
47 00:01:16,959 –> 00:01:18,750 of all of these new symbols.
48 00:01:19,370 –> 00:01:20,709 However, if you watch my
49 00:01:20,720 –> 00:01:21,919 explanations for the set
50 00:01:21,930 –> 00:01:23,430 theory, you already know
51 00:01:23,489 –> 00:01:25,139 that this here is not a
52 00:01:25,150 –> 00:01:26,400 complete definition for a
53 00:01:26,410 –> 00:01:26,870 set.
54 00:01:27,389 –> 00:01:28,430 Therefore, the question would
55 00:01:28,440 –> 00:01:30,139 be how do we define the
56 00:01:30,150 –> 00:01:31,779 natural numbers just with
57 00:01:31,790 –> 00:01:32,279 sets?
58 00:01:32,849 –> 00:01:34,089 This is what we’ll answer
59 00:01:34,099 –> 00:01:35,029 in this video.
60 00:01:35,040 –> 00:01:36,629 More precisely, we will
61 00:01:36,639 –> 00:01:38,360 construct the natural numbers
62 00:01:38,370 –> 00:01:40,010 that also include zero as
63 00:01:40,019 –> 00:01:40,589 a number.
64 00:01:41,190 –> 00:01:42,940 I will use the symbol N zero
65 00:01:42,949 –> 00:01:44,930 for this set but be careful
66 00:01:44,940 –> 00:01:45,519 often.
67 00:01:45,529 –> 00:01:47,410 Also N is used for this
68 00:01:47,419 –> 00:01:47,889 set here.
69 00:01:48,680 –> 00:01:49,089 OK.
70 00:01:49,099 –> 00:01:50,489 With this, let’s start
71 00:01:50,500 –> 00:01:51,910 constructing the natural
72 00:01:51,919 –> 00:01:52,599 numbers.
73 00:01:53,209 –> 00:01:55,000 As we have seen in our examples
74 00:01:55,010 –> 00:01:56,599 with apples and oranges.
75 00:01:56,610 –> 00:01:58,449 The size of a set can be
76 00:01:58,459 –> 00:02:00,019 used to define a number
77 00:02:00,690 –> 00:02:02,489 and we already know one particular
78 00:02:02,500 –> 00:02:04,199 set, namely the empty
79 00:02:04,209 –> 00:02:06,150 set by definition,
80 00:02:06,160 –> 00:02:07,669 the empty set does not have
81 00:02:07,680 –> 00:02:08,750 any elements.
82 00:02:08,758 –> 00:02:10,169 So it represents the number
83 00:02:10,179 –> 00:02:10,669 zero.
84 00:02:11,470 –> 00:02:12,830 And now it makes sense to
85 00:02:12,839 –> 00:02:14,729 use this as a definition
86 00:02:14,740 –> 00:02:16,490 for our new symbol zero.
87 00:02:17,080 –> 00:02:18,029 Maybe it looks a little bit
88 00:02:18,039 –> 00:02:18,759 strange here.
89 00:02:18,770 –> 00:02:20,369 But keep in mind we already
90 00:02:20,380 –> 00:02:21,809 have the set theory.
91 00:02:21,820 –> 00:02:23,449 So it makes sense that every
92 00:02:23,460 –> 00:02:25,449 new object we introduce should
93 00:02:25,460 –> 00:02:26,490 also be a set.
94 00:02:27,210 –> 00:02:28,449 We did the same thing when
95 00:02:28,460 –> 00:02:30,339 we introduced ordered pairs
96 00:02:30,350 –> 00:02:31,220 and maps.
97 00:02:31,839 –> 00:02:33,179 Now for the next step for
98 00:02:33,190 –> 00:02:34,789 defining the number one as
99 00:02:34,800 –> 00:02:36,419 a set, we need a new
100 00:02:36,429 –> 00:02:38,229 set that contains only one
101 00:02:38,240 –> 00:02:40,210 element we could take
102 00:02:40,220 –> 00:02:41,830 any element we want.
103 00:02:41,910 –> 00:02:43,009 But the only thing that is
104 00:02:43,020 –> 00:02:44,729 defined yet is the empty set.
105 00:02:45,800 –> 00:02:47,619 So we put zero inside the
106 00:02:47,630 –> 00:02:49,550 set brackets and we get out
107 00:02:49,559 –> 00:02:50,229 a new set.
108 00:02:50,889 –> 00:02:52,229 What you should see is that
109 00:02:52,240 –> 00:02:54,190 what we do here is exactly
110 00:02:54,199 –> 00:02:55,479 the same thing we had in
111 00:02:55,490 –> 00:02:56,789 mind when we dealt with the
112 00:02:56,800 –> 00:02:57,550 oranges.
113 00:02:58,000 –> 00:02:59,270 However, here we don’t need
114 00:02:59,279 –> 00:03:00,949 any real world objects.
115 00:03:00,960 –> 00:03:02,589 We just need the abstract
116 00:03:02,600 –> 00:03:04,300 concepts in set theory.
117 00:03:04,899 –> 00:03:05,240 OK.
118 00:03:05,250 –> 00:03:06,470 Now, for the number two,
119 00:03:06,479 –> 00:03:07,949 we need two elements.
120 00:03:07,960 –> 00:03:09,050 So what do we do?
121 00:03:09,080 –> 00:03:10,990 We take the only two elements
122 00:03:11,000 –> 00:03:11,669 we know.
123 00:03:12,470 –> 00:03:14,009 So zero and one,
124 00:03:14,020 –> 00:03:15,360 because we know they are
125 00:03:15,369 –> 00:03:17,350 different at this point,
126 00:03:17,360 –> 00:03:18,589 you should see how this game
127 00:03:18,600 –> 00:03:20,479 works and we can just continue
128 00:03:20,490 –> 00:03:20,639 it.
129 00:03:21,369 –> 00:03:23,059 Therefore, this is our number
130 00:03:23,070 –> 00:03:25,059 three as a set in the same
131 00:03:25,070 –> 00:03:26,220 way, we can write down the
132 00:03:26,229 –> 00:03:27,139 number four.
133 00:03:27,699 –> 00:03:29,380 However, here you might already
134 00:03:29,389 –> 00:03:30,880 wish for a shorter way to
135 00:03:30,889 –> 00:03:31,679 write it down.
136 00:03:32,399 –> 00:03:33,860 And of course, if you compare
137 00:03:33,869 –> 00:03:35,690 to the set before three,
138 00:03:35,699 –> 00:03:37,619 we just add one element.
139 00:03:38,089 –> 00:03:39,820 For this reason, we can just
140 00:03:39,830 –> 00:03:41,179 write it as a union.
141 00:03:42,000 –> 00:03:43,690 So this is what it is and
142 00:03:43,699 –> 00:03:45,229 maybe it looks again a little
143 00:03:45,240 –> 00:03:46,500 bit strange, but keep in
144 00:03:46,509 –> 00:03:48,009 mind three is a set
145 00:03:48,020 –> 00:03:48,529 here.
146 00:03:48,699 –> 00:03:49,889 And here we have the set
147 00:03:49,899 –> 00:03:51,410 that contains three as an
148 00:03:51,419 –> 00:03:52,050 element.
149 00:03:52,619 –> 00:03:52,949 OK?
150 00:03:52,960 –> 00:03:54,410 But with this formula, you
151 00:03:54,419 –> 00:03:56,130 now know how to construct
152 00:03:56,139 –> 00:03:57,800 the next number in order.
153 00:03:58,550 –> 00:04:00,070 And then we get all the
154 00:04:00,080 –> 00:04:01,770 numbers we want to have.
155 00:04:02,300 –> 00:04:03,529 And after that, we want to
156 00:04:03,539 –> 00:04:05,309 put them into one set
157 00:04:05,440 –> 00:04:07,240 and call it the natural numbers
158 00:04:07,250 –> 00:04:07,979 and zero.
159 00:04:08,610 –> 00:04:09,970 However, there you might
160 00:04:09,979 –> 00:04:11,630 see a problem because this
161 00:04:11,639 –> 00:04:13,479 whole construction here never
162 00:04:13,490 –> 00:04:14,089 stops.
163 00:04:14,720 –> 00:04:15,360 Or to put it.
164 00:04:15,369 –> 00:04:16,670 In other words, the natural
165 00:04:16,678 –> 00:04:18,608 numbers, the set and zero
166 00:04:18,619 –> 00:04:20,420 should have infinitely many
167 00:04:20,428 –> 00:04:21,170 elements.
168 00:04:21,910 –> 00:04:23,010 And that is something we
169 00:04:23,019 –> 00:04:24,660 have to put into our set
170 00:04:24,670 –> 00:04:25,209 theory.
171 00:04:25,220 –> 00:04:27,040 So we call it an axiom.
172 00:04:27,700 –> 00:04:28,920 It’s simply something we
173 00:04:28,929 –> 00:04:30,799 put as a true statement
174 00:04:30,809 –> 00:04:32,059 into our theory.
175 00:04:32,480 –> 00:04:34,160 For example, the existence
176 00:04:34,170 –> 00:04:35,880 of the empty set was also
177 00:04:35,890 –> 00:04:36,619 an axiom.
178 00:04:36,630 –> 00:04:38,459 We put into the set theory.
179 00:04:38,920 –> 00:04:40,429 And now we want to put in
180 00:04:40,440 –> 00:04:41,899 the existence of the natural
181 00:04:41,910 –> 00:04:42,459 numbers.
182 00:04:43,019 –> 00:04:44,850 So we say there’s a set N
183 00:04:44,859 –> 00:04:46,570 zero with the following two
184 00:04:46,579 –> 00:04:47,380 properties.
185 00:04:47,950 –> 00:04:49,160 The first one is that we
186 00:04:49,170 –> 00:04:50,239 have a starting point at
187 00:04:50,250 –> 00:04:50,799 zero.
188 00:04:50,829 –> 00:04:52,119 So the set zero
189 00:04:52,130 –> 00:04:53,859 lies in N zero.
190 00:04:54,549 –> 00:04:55,700 And the second one tells
191 00:04:55,709 –> 00:04:57,470 us that for all numbers in
192 00:04:57,480 –> 00:04:59,410 N zero, also the
193 00:04:59,420 –> 00:05:01,290 successor lies in N zero.
194 00:05:01,809 –> 00:05:03,140 And now we know we can write
195 00:05:03,149 –> 00:05:04,559 it as this set here
196 00:05:04,899 –> 00:05:06,579 as a reminder here, we have
197 00:05:06,589 –> 00:05:08,250 the conditional form logic.
198 00:05:08,279 –> 00:05:10,230 And I already told you sometimes
199 00:05:10,239 –> 00:05:11,290 it’s written with a double
200 00:05:11,299 –> 00:05:13,059 arrow later, we will
201 00:05:13,070 –> 00:05:14,769 also do this just to avoid
202 00:05:14,779 –> 00:05:16,140 confusion with the arrow
203 00:05:16,149 –> 00:05:16,970 for maps.
204 00:05:17,649 –> 00:05:19,140 However, to finish our axiom
205 00:05:19,149 –> 00:05:20,660 here, we also have to say
206 00:05:20,670 –> 00:05:22,029 that N zero is the
207 00:05:22,040 –> 00:05:23,730 smallest set that has
208 00:05:23,739 –> 00:05:24,779 these properties.
209 00:05:25,269 –> 00:05:26,769 This simply means that any
210 00:05:26,779 –> 00:05:28,019 other set with these two
211 00:05:28,029 –> 00:05:29,880 properties is a superset
212 00:05:29,890 –> 00:05:30,899 of N zero.
213 00:05:31,339 –> 00:05:31,890 OK.
214 00:05:31,899 –> 00:05:33,500 And with this, we finally
215 00:05:33,510 –> 00:05:34,769 have the natural numbers.
216 00:05:34,779 –> 00:05:35,989 In our set theory,
217 00:05:36,700 –> 00:05:38,279 it’s constructed with sets,
218 00:05:38,290 –> 00:05:39,880 it has sets as elements
219 00:05:39,890 –> 00:05:41,279 such that we can work with
220 00:05:41,290 –> 00:05:41,440 it.
221 00:05:42,130 –> 00:05:43,040 That’s how you should see
222 00:05:43,049 –> 00:05:43,279 it.
223 00:05:43,290 –> 00:05:44,559 It’s not the explanation
224 00:05:44,570 –> 00:05:45,480 of numbers.
225 00:05:45,510 –> 00:05:47,079 It’s a working tool for us.
226 00:05:47,839 –> 00:05:49,570 Therefore, it’s often called
227 00:05:49,579 –> 00:05:51,450 a model for the natural numbers.
228 00:05:52,109 –> 00:05:53,290 Now, with the natural numbers,
229 00:05:53,299 –> 00:05:54,609 we immediately get a map,
230 00:05:54,619 –> 00:05:56,209 we call the successor map,
231 00:05:57,209 –> 00:05:58,790 we denote it with S and it
232 00:05:58,799 –> 00:06:00,649 goes from N zero into
233 00:06:00,660 –> 00:06:01,390 N zero.
234 00:06:01,850 –> 00:06:03,149 And the definition, we already
235 00:06:03,160 –> 00:06:04,869 know we send the number X
236 00:06:04,929 –> 00:06:05,869 to the union
237 00:06:06,760 –> 00:06:08,640 because we learned this gives
238 00:06:08,649 –> 00:06:09,760 us the successor.
239 00:06:10,350 –> 00:06:11,899 For example, if we put the
240 00:06:11,910 –> 00:06:13,540 number six into the map,
241 00:06:13,549 –> 00:06:15,260 we get out the number seven.
242 00:06:16,130 –> 00:06:16,529 OK?
243 00:06:16,540 –> 00:06:17,670 I think that’s good enough
244 00:06:17,679 –> 00:06:18,470 for an introduction.
245 00:06:19,049 –> 00:06:20,100 In the next video, we will
246 00:06:20,109 –> 00:06:21,510 talk about all the properties
247 00:06:21,519 –> 00:06:22,989 of the natural numbers and
248 00:06:23,000 –> 00:06:24,850 also how we can define the
249 00:06:24,859 –> 00:06:25,470 addition.
250 00:06:26,000 –> 00:06:27,089 So I hope I see you there
251 00:06:27,100 –> 00:06:28,190 and have a nice day.
252 00:06:28,279 –> 00:06:28,980 Bye.
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Quiz Content
Q1: By using sets, one can define the natural numbers $\mathbb{N}_0$. How is the number $0$ defined?
A1: $ 0 = \emptyset$
A2: $ 0 = { \emptyset, \emptyset } $
A3: $ 0 = { 1} $
A4: $ 0 = {1,2} $
Q2: Formally, the set $\mathbb{N}_0$ is introduced axiomatically. What is not a property of this set?
A1: For all $x$, we have that $x \in \mathbb{N}_0$ implies $ x \cap {x} \in \mathbb{N}_0$.
A2: $0 \in \mathbb{N}_0$.
A3: For all $x$, we have that $x \in \mathbb{N}_0$ implies $ x \cup {x} \in \mathbb{N}_0$.
Q3: The natural numbers have the important successor map $s$. What is $s(0)$?
A1: $1$
A2: $0$
A3: $2$
A4: $3$
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Last update: 2024-10
