def setup(self):
self.shift = 1.1
shift = self.shift
self.omega1 = Omega1()
self.omega1.shift(shift * UP)
self.omega1_brace = BraceDesc(self.omega1, "\\omega_1", UP)
self.add(self.omega1, self.omega1_brace)
# countable subset
omega = OrdinalOmega(x1=-2)
omega.stretch(0.5, 1)
omega.next_to(self.omega1[1])
self.count_subset = VGroup(
self.omega1[1][0],
LimitSubOrdinal([subord[0] for subord in self.omega1[1][1:]]),
).copy()
self.count_subset.add(omega)
self.al_count_subset = VGroup(self.omega1[1][:3]).copy()
self.shifted_count_subset = self.count_subset.copy()
VGroup(self.al_count_subset,
self.shifted_count_subset).shift(2 * shift * DOWN)
omega.stretch_in_place(0, 1)
self.count_brace = BraceText(self.al_count_subset, "spočetná")
def setup(self):
self.shift = 1.1
shift = self.shift
self.omega = OrdinalOmega()
self.omega_brace = BraceDesc(self.omega, "\\omega", UP)
self.omega.set_color(BLUE)
self.add(self.omega, self.omega_brace)
# finite subset
self.subset_fin = VGroup(self.omega[2:30:3]).copy()
self.al_subset_fin = VGroup(self.omega[:len(self.subset_fin)]).copy()
self.shifted_subset_fin = self.subset_fin.copy()
VGroup(self.al_subset_fin, self.shifted_subset_fin).shift(shift * DOWN)
self.brace_fin = BraceText(self.al_subset_fin, "konečná")
# infinite subset
self.subset_inf = LimitSubOrdinal(
[bar for i, bar in enumerate(self.omega) if is_prime(i)]).copy()
self.al_subset_inf = self.omega.copy()
self.shifted_subset_inf = self.subset_inf.copy()
self.subset_inf.shift(shift * UP)
VGroup(self.al_subset_inf, self.shifted_subset_inf).shift(shift * DOWN)
self.brace_inf = BraceText(self.al_subset_inf, "stejná")
def construct(self):
omega = OrdinalOmega()
omega_brace = BraceDesc(omega, "\\omega", UP)
self.add(omega_brace)
omega, _ = self.play_infinite_supremum(waits=(4, 5.1))
fin_index = 12
bar = omega[fin_index].copy()
bar.highlight(YELLOW)
arrow = Arrow(
bar.get_edge_center(DOWN) + DOWN,
bar.get_edge_center(DOWN),
)
label = TextMobject("konečný ordinál").next_to(arrow, DOWN)
self.wait_to(16)
self.play(
ShowCreation(bar),
ShowCreation(arrow),
FadeIn(label),
)
finite_brace = Brace(VGroup(omega[:fin_index]), DOWN)
brace_label = label.copy()
finite_brace.put_at_tip(brace_label)
self.wait_to(20.3)
self.play(
GrowFromCenter(finite_brace),
ReplacementTransform(label, brace_label),
)
terminal_segment_brace = BraceText(
Line(omega[fin_index + 1].get_center() + DOWN,
omega.get_edge_center(RIGHT)), "nekonečno")
self.wait_to(25.5)
self.play(terminal_segment_brace.creation_anim())
self.wait_to(41)
self.play(
FadeOut(
VGroup(
finite_brace,
brace_label,
terminal_segment_brace,
bar,
arrow,
)))
def construct(self):
negatives = VGroup(*[self.number_tex(-n) for n in range(9)])
positives = VGroup(*[self.number_tex(n) for n in range(9)])
negatives.shift(UP)
neg_brace = BraceDesc(negatives, "\\aleph_0", UP)
pos_brace = BraceDesc(positives, "\\aleph_0", DOWN)
self.play(ShowCreation(negatives))
self.play(neg_brace.creation_anim())
self.dither()
self.play(ShowCreation(positives))
self.play(pos_brace.creation_anim())
self.dither()
self.play(
positives.shift,
UP,
pos_brace.shift,
UP,
)
self.remove(positives[0])
self.play(
FadeOut(pos_brace),
neg_brace.shift_brace,
VGroup(negatives, negatives[-1].copy().move_to(positives[-1])),
)
self.dither()
return
conversation = Conversation(self)
conversation.add_bubble("Hey, what happened to the merged zero?")
self.dither()
conversation.add_bubble("The size is still countable infinite.")
self.dither()
def construct(self):
#self.force_skipping()
self.cur_color = WHITE
self.number_mul = 1
self.make_numbers(num_num=14)
self.numbers.to_corner(UP + LEFT)
self.numbers.shift(1.5 * RIGHT)
self.add(self.numbers)
even, _, even_spaced, even_arranged = self.make_subset(
lambda n: 2 * n, 3)
odd, _, odd_spaced, odd_arranged = self.make_subset(
lambda n: 2 * n + 1, 2)
odd_arranged.highlight(YELLOW)
even_arranged.highlight(RED)
self.play(
Transform(even, even_spaced),
Transform(odd, odd_spaced),
)
self.play(
Transform(even, even_arranged),
Transform(odd, odd_arranged),
)
self.dither()
colors = color_gradient([YELLOW, GREEN, BLUE], 6)
ori_numbers = self.numbers
self.numbers = even
self.number_mul = 2
self.cur_color = RED
table = [odd]
for index, color in enumerate(colors[1:]):
subodd, _, _, subodd_arranged \
= self.make_subset(lambda n: 2*n+1, 0)
table += [subodd]
subeven, _, subeven_spaced, subeven_arranged \
= self.make_subset(lambda n: 2*n, 1)
self.remove(self.numbers)
self.add(subodd, subeven)
subodd_arranged.highlight(color)
self.play(Transform(subeven, subeven_spaced))
if False and index < 2:
self.play(Transform(subeven, subeven_arranged))
self.dither()
self.play(Transform(subodd, subodd_arranged))
self.dither()
else:
self.play(
Transform(subeven, subeven_arranged),
Transform(subodd, subodd_arranged),
)
self.number_mul *= 2
self.numbers = subeven
missing_zero = ori_numbers[0].copy().highlight(RED)
self.play(Write(missing_zero))
self.remove(ori_numbers[0])
ori_numbers.save_state()
ori_numbers.remove(ori_numbers[0])
self.play(missing_zero.behind_edge, DOWN)
self.dither()
h_brace = BraceDesc(odd, "\\aleph_0", UP)
h_brace.desc.highlight(YELLOW)
v_brace = BraceDesc(VGroup(odd, subeven), "\\aleph_0", LEFT)
v_brace.desc.highlight(BLUE)
v_brace.shift(0.2 * LEFT)
self.play(
FadeOut(ori_numbers),
h_brace.creation_anim(),
)
self.dither()
self.play(v_brace.creation_anim())
self.dither()
prod_formula = TexMobject("\\aleph_0", "\\cdot", "\\aleph_0",
" = \\aleph_0")
prod_formula[0].highlight(BLUE)
prod_formula[2].highlight(YELLOW)
prod_formula.to_corner(LEFT + UP)
self.play(
ReplacementTransform(v_brace.desc[0].copy(), prod_formula[0]),
ReplacementTransform(h_brace.desc[0].copy(), prod_formula[2]),
)
self.play(Write(VGroup(prod_formula[1], prod_formula[3])))
self.dither()
table_but_first_column = VGroup(*[VGroup(*row[1:]) for row in table])
self.play(
FadeOut(VGroup(h_brace, v_brace, prod_formula)),
table_but_first_column.set_fill,
None,
0.4,
)
formula_r_ori = None
for i, row in enumerate(table[:4]):
formula_r = TexMobject("2^{" + str(i) + "}=")
formula_r.highlight(
rgb_to_color(row[0].family_members_with_points()[0].fill_rgb))
formula_r.next_to(row[0], LEFT, buff=0.3, aligned_edge=DOWN)
if formula_r_ori is None: self.play(Write(formula_r))
else: self.play(ReplacementTransform(formula_r_ori, formula_r))
self.dither()
formula_r_ori = formula_r
formula_r = TexMobject("2^r=")
formula_r.shift(formula_r_ori[0].get_center() -
formula_r[0].get_center())
formula_r.highlight(BLUE)
formula_r[-1].set_fill(opacity=0)
formula_r.to_edge(DOWN)
self.play(ReplacementTransform(formula_r_ori, formula_r))
self.dither()
self.play(table_but_first_column.set_fill, None, 1)
table_but_first_row = VGroup(*table[1:])
self.play(table_but_first_row.set_fill, None, 0.4)
self.dither()
formula_c = TexMobject("(2c+1)")
formula_c.to_edge(UP)
formula_c.highlight(YELLOW)
self.play(Write(formula_c))
self.dither()
self.play(table_but_first_row.set_fill, None, 1)
show_r = 3
show_c = 5
show_circ = Circle(radius=0.45)
show_circ.highlight(GREEN)
show_circ.move_to(table[show_r][show_c])
show_r_circ = show_circ.copy()
show_r_circ.move_to(table[show_r][0], coor_mask=X_MASK)
show_r_circ.highlight(BLUE)
show_c_circ = show_circ.copy()
show_c_circ.move_to(table[0][show_c], coor_mask=Y_MASK)
show_c_circ.highlight(YELLOW)
self.play(ShowCreation(show_circ))
self.play(ShowCreation(show_c_circ))
self.play(ShowCreation(show_r_circ))
self.dither()
formula_prod = TexMobject("2^r", '\\cdot', "(2c+1)", "-1")
formula_prod.to_corner(UP + LEFT)
formula_prod[0].highlight(BLUE)
formula_prod[2].highlight(YELLOW)
self.play(
FadeIn(formula_prod[1]),
ReplacementTransform(formula_r.copy(), formula_prod[0]),
ReplacementTransform(formula_c.copy(), formula_prod[2]),
)
self.dither()
self.play(
FadeOut(
VGroup(formula_r, formula_c, show_circ, show_c_circ,
show_r_circ)))
self.dither()
pairs_table = VGroup(*[
VGroup(*[self.pair_tex(r, c, mob) for c, mob in enumerate(row)])
for r, row in enumerate(table)
])
table = VGroup(*table)
for row in table:
for num in row:
num.submobjects = [VGroup(digit) for digit in num.submobjects]
self.revert_to_original_skipping_status()
self.dither()
self.play(ReplacementTransform(table, pairs_table))
self.dither()
h_shift = RIGHT * (table[0][1].get_center() - table[0][0].get_center())
table = pairs_table
table_dest = table.copy()
for r, row in enumerate(table_dest):
for c, num in enumerate(row):
num.shift(h_shift * ((2**r) * (2 * c + 1) - 1 - c))
self.play(Transform(table, table_dest))
table_dest = table.copy()
for row in table_dest:
for num in row:
num.move_to(2 * DOWN, coor_mask=Y_MASK)
self.play(Transform(table, table_dest))
self.dither()
pairs_row = VGroup(*it.chain(*table))
pairs_row.submobjects.sort(key=lambda mob: mob.get_center()[0])
ori_numbers.restore()
ori_numbers.shift(pairs_row[0].get_center() -
ori_numbers[1].get_center())
ori_numbers.shift(2 * UP)
self.add(ori_numbers)
matching = VGroup(*[
Line(num.get_edge_center(DOWN), pair.get_edge_center(UP), buff=0.2)
for num, pair in zip(ori_numbers[1:], pairs_row)
])
self.play(
ShowCreation(ori_numbers),
ShowCreation(matching),
)
self.dither()
matching.add(matching[-1].copy())
matching[-1].shift(h_shift)
general_pair = TexMobject("r,c")
general_pair[0].highlight(BLUE)
general_pair[2].highlight(YELLOW)
visible_formula_prod = VGroup(*formula_prod[:3])
general_pair.move_to(visible_formula_prod)
general_pair.shift(2 * DOWN)
general_matching = Line(general_pair.get_edge_center(UP),
visible_formula_prod.get_edge_center(DOWN),
buff=0.2)
self.play(FadeIn(general_pair))
self.play(ShowCreation(general_matching))
self.dither()
self.play(Write(formula_prod[-1]))
self.play(VGroup(matching, pairs_row).shift, -h_shift)
self.dither()
def construct(self):
#self.force_skipping()
self.make_numbers()
self.numbers.to_corner(UP + LEFT)
even, even_ori, even_spaced, even_arranged = self.make_subset(
lambda n: 2 * n, 2)
odd, odd_ori, odd_spaced, odd_arranged = self.make_subset(
lambda n: 2 * n + 1, 3)
self.add(self.numbers)
self.play(Transform(even, even_spaced))
VGroup(even_ori, even_spaced, even_arranged).highlight(RED)
self.play(Transform(even, even_arranged))
self.dither()
self.play(Transform(odd, odd_spaced))
VGroup(odd_ori, odd_spaced, odd_arranged).highlight(YELLOW)
self.play(Transform(odd, odd_arranged))
self.dither()
brace = BraceDesc(odd, "\\aleph_0")
self.play(brace.creation_anim())
self.dither()
self.play(
Transform(even, even_spaced),
Transform(odd, odd_spaced),
)
self.play(
Transform(even, even_ori),
Transform(odd, odd_ori),
)
self.remove(self.numbers)
self.dither()
self.play(
Transform(even, even_spaced),
Transform(odd, odd_spaced),
)
self.play(
Transform(even, even_arranged),
Transform(odd, odd_arranged),
)
self.dither()
even_abstr = VGroup(
*[Dot(num.get_center(), color=RED) for num in even])
odd_abstr = VGroup(
*[Dot(num.get_center(), color=YELLOW) for num in odd])
self.play(
Transform(even, even_abstr),
Transform(odd, odd_abstr),
)
self.dither()
apples_shrinked, apples, apples_spaced, apples_in_set \
= self.make_fruit_set(Apple().scale(0.6),
even_arranged, even_spaced, even_ori)
pears_shrinked, pears, pears_spaced, pears_in_set \
= self.make_fruit_set(Pear().scale(0.6),
odd_arranged, odd_spaced, odd_ori)
self.play(
ReplacementTransform(apples_shrinked, apples),
ReplacementTransform(pears_shrinked, pears),
)
self.remove(even, odd)
self.dither()
self.play(
Transform(apples, apples_spaced),
Transform(pears, pears_spaced),
)
self.play(
Transform(apples, apples_in_set),
Transform(pears, pears_in_set),
)
self.dither()
sum_formula = TexMobject('\\aleph_0', '+', '\\aleph_0', "=\\aleph_0")
sum_formula[0].highlight(RED)
sum_formula[2].highlight(YELLOW)
sum_formula.shift(BOTTOM / 2)
self.play(Write(sum_formula))
self.dither()
def construct(self):
self.h_space = 2
self.colors = color_gradient(
[YELLOW, GREEN, BLUE, DARK_BLUE, DARK_GRAY], 20)
numbers = self.make_numbers()
self.play(
ShowCreation(numbers[0]),
ShowCreation(numbers[1]),
)
brace = BraceDesc(numbers[0], "\\aleph_0", DOWN)
brace.next_to(numbers[1], DOWN, coor_mask=Y_MASK, buff=0.3)
self.play(brace.creation_anim())
skip = False
denominator = 2
parts = []
while True:
parts.append(numbers[0].copy())
self.add(parts[-1])
self.play(VGroup(numbers, brace).shift, 0.9 * DOWN)
self.remove(numbers)
if denominator == 6: break
numbers = self.make_numbers(denominator, numbers)
ori_numbers = self.make_numbers(denominator - 1, numbers)
if skip:
offset = (len(ori_numbers[1]) // 2 + 1) % 2
for label in ori_numbers[1][offset::2]:
label.set_fill(opacity=0)
if denominator == 4: skip = True
if skip:
offset = (len(numbers[1]) // 2 + 1) % 2
for label in numbers[1][offset::2]:
label.set_fill(opacity=0)
self.play(ReplacementTransform(ori_numbers, numbers))
denominator += 1
self.dither()
self.remove(brace, numbers)
while denominator < len(self.colors) + 1:
part = self.make_dots(denominator)
part.move_to(parts[-1])
part.shift(0.5 * DOWN)
parts.append(part)
denominator += 1
parts.reverse()
parts = VGroup(*parts)
parts_dest = parts.copy()
for part in parts_dest:
part.center()
self.play(Transform(parts, parts_dest), run_time=2)
self.dither()
rationals = TexMobject('|', "\\text{Rational Numbers}",
"| = \\aleph_0")
rationals.shift(0.5 * DOWN - rationals[1].get_edge_center(UP))
self.play(Write(rationals[1]))
self.dither()
dots7 = self.make_dots(7)
dots7.shift(2 * DOWN)
center = len(dots7) // 2
pi_dot = dots7[center + 22]
pi_label = TexMobject("\\frac{22}{7}")
pi_label.next_to(pi_dot, DOWN)
self.play(Write(pi_label))
self.dither()
self.play(ShowCreation(dots7))
self.play(FocusOn2(pi_dot, scale=3))
dots7.remove(pi_dot)
self.play(FadeOut(dots7))
self.play(VGroup(pi_dot, pi_label).shift, 2 * UP)
self.dither()
self.play(FadeOut(VGroup(pi_dot, pi_label)))
self.play(Write(VGroup(rationals[0], rationals[2])))
conversation1 = Conversation(self)
conversation1.add_bubble("Are all sets countable?")
self.dither()
conversation2 = Conversation(self, start_answer=True)
conversation2.add_bubble("Of course not.")
self.dither()
self.prepare_overall_picture()
self.overall_picture.to_edge(UP, buff=0.4)
self.overall_picture.remove(self.uncountable_brace)
self.play(FadeIn(self.overall_picture))
self.dither()
self.play(self.uncountable_brace.creation_anim())
self.dither()
real_size = TexMobject("|\\mathbb R|")
real_size.set_color(GREEN)
real_size.next_to(self.sizes, buff=1)
self.play(Write(real_size))
self.dither()
def construct(self):
self.prepare_overall_picture()
self.subtitle.to_edge(UP)
self.add(self.subtitle)
self.overall_picture.remove(self.finite_brace, self.infinite_brace)
self.overall_picture.next_to(self.subtitle,
DOWN,
buff=0.5,
coor_mask=Y_MASK)
real_size = TexMobject("|\\mathbb R|")
real_size.set_color(GREEN)
real_size.next_to(self.sizes, buff=1)
self.sizes.add(real_size)
self.play(FadeIn(self.sizes))
self.play(self.countable_brace.creation_anim())
self.play(self.uncountable_brace.creation_anim())
ordinal_bg = OrdinalFiniteProd(OrdinalOmega, 3, x1=8)
ordinal_bg.shift(2 * DOWN)
ordinals_title = TextMobject("Ordinály")
ordinals_title.next_to(ordinal_bg, UP, aligned_edge=LEFT, buff=0.3)
ordinal = VGroup(ordinal_bg[0], VGroup(*ordinal_bg[1][:3])).copy()
ordinal_bg.set_color(DARK_GREY)
self.wait_to(3.7)
self.play(FadeIn(ordinal_bg), FadeIn(ordinals_title))
self.play(ShowCreation(ordinal[0]))
self.play(ShowCreation(ordinal[1]))
self.dither()
self.wait_to(11.5)
self.play(FadeOut(self.overall_picture))
card_sum = TexMobject("\\aleph_0 + \\aleph_0 = \\aleph_0")
card_prod = TexMobject("\\aleph_0\\cdot\\aleph_0 = \\aleph_0")
card_sum.next_to(self.subtitle, DOWN)
card_prod.next_to(card_sum, DOWN)
card_prod.shift(
(card_sum[-1].get_center() - card_prod[-1].get_center()) * X_MASK)
self.play(Write(card_sum))
self.wait_to(17.7)
self.play(Write(card_prod))
ordinal_dest = ordinal.copy()
ordinal_bg_dest = ordinal_bg.copy()
ordinals_title_dest = ordinals_title.copy()
VGroup(ordinal_bg_dest, ordinal_dest,
ordinals_title_dest).shift(0.7 * UP)
ord_type_brace = BraceDesc(ordinal_dest, "\\omega+3")
self.wait_to(22)
self.play(
Transform(ordinal_bg, ordinal_bg_dest),
Transform(ordinal, ordinal_dest),
Transform(ordinals_title, ordinals_title_dest),
ord_type_brace.creation_anim(),
)
bar = ordinal_bg[1][3].copy()
bar.set_color(BLUE)
pointer = TrianglePointer(color=BLUE).next_to(bar, UP)
index_label = ord_type_brace.desc.copy()
index_label.set_color(BLUE)
index_label.next_to(pointer, UP)
self.wait_to(29.5)
self.play(
ShowCreation(bar),
FadeIn(pointer),
FadeIn(index_label),
)
self.wait_to(36.5)
def construct(self):
#self.force_skipping()
supremum = Supremum.construct(self, fast_play=True) # 2
self.wait_to(3)
omega1_brace = BraceDesc(supremum, "\\omega_1", UP)
self.play(omega1_brace.creation_anim())
omega1 = Omega1()
omega1_src = omega1.copy()
omega1_src[0].highlight(BLACK)
omega1_src[1].next_to(supremum, buff=0)
omega1_src[1].add_to_back(*supremum.submobjects)
self.wait_to(6.5)
self.play(
ReplacementTransform(omega1_src, omega1),
omega1_brace.shift_brace,
omega1,
)
self.revert_to_original_skipping_status()
countable_point = Dot(2 * RIGHT)
countable_point.set_color(YELLOW)
countable_arrow = Arrow(
countable_point.get_center() + DOWN,
countable_point.get_center(),
)
countable_label = TextMobject("spočetný ordinál")
countable_label.next_to(countable_arrow, DOWN)
brace = Brace(
Line(omega1.get_corner(LEFT + DOWN), countable_point.get_center()),
DOWN)
brace_label = countable_label.copy()
brace.put_at_tip(brace_label)
self.wait_to(9)
self.play(
FadeIn(countable_label),
FadeIn(countable_point),
ShowCreation(countable_arrow),
)
self.wait_to(11.5)
self.play(
GrowFromCenter(brace),
ReplacementTransform(countable_label, brace_label),
)
self.wait_to(24.5)
uncountable_brace = Brace(omega1, DOWN)
uncountable_label = TextMobject("nespočetný ordinál")
uncountable_brace.put_at_tip(uncountable_label)
brace_label.add_to_back(brace_label[0].copy(), brace_label[0].copy())
self.play(
Transform(brace, uncountable_brace),
Transform(brace_label, uncountable_label),
)
self.remove(brace_label)
brace_label = TextMobject("nespočetný", "ordinál")
brace.put_at_tip(brace_label)
self.add(brace_label)
uncountable_brace = Brace(
Line(countable_point.get_center() + DOWN,
omega1[0].get_edge_center(RIGHT)), DOWN)
uncountable_label = TextMobject("nespočetno", "ordinál")
uncountable_brace.put_at_tip(uncountable_label)
uncountable_label[1].highlight(BLACK)
uncountable_label.shift(uncountable_label.get_center() -
uncountable_label[0].get_center())
self.wait_to(46)
self.play(
Transform(brace, uncountable_brace),
Transform(brace_label, uncountable_label),
)
self.wait_to(56)
omega = OrdinalOmega()
self.play(
FadeOut(
VGroup(
omega1,
brace,
brace_label,
countable_point,
countable_arrow,
)),
omega1_brace.change_brace_desc,
omega,
"\\omega",
)
def construct(self, fast_play=False):
#self.force_skipping()
omega = OrdinalOmega()
omega_plus_5 = OrdinalSum(OrdinalOmega, 0.75,
lambda **kwargs: OrdinalFinite(5, **kwargs))
omega_times_3 = OrdinalFiniteProd(OrdinalOmega, 3)
omega_pow_3 = make_ordinal_power(3, q=(0.7, 0.84, 0.84))
ordinals = VGroup(omega, omega_plus_5, omega_times_3, omega_pow_3)
ordinals.stretch(0.4, 1)
ordinals.arrange_submobjects(DOWN, aligned_edge=LEFT, center=False)
ordinals.shift(-Y_MASK * ordinals.get_center())
dots = TexMobject("\\vdots")
dots.next_to(ordinals, DOWN)
if fast_play:
self.add(ordinals, dots)
else:
self.play(ShowCreation(omega))
self.wait_to(6)
self.play(ShowCreation(omega_plus_5))
self.wait_to(7.3)
self.play(ShowCreation(omega_times_3))
self.wait_to(8.7)
self.play(ShowCreation(omega_pow_3))
self.play(FadeIn(dots))
result = OrdinalFiniteProd(OrdinalOmega, 5, x1=8)
result_stretched = result.copy()
result_stretched.stretch(
omega[0].get_height() / result_stretched[0][0].get_height(), 1)
al_omega = result_stretched[0].copy()
al_omega.shift(omega[0].get_center() - al_omega[0].get_center())
al_omega_plus_5 = VGroup(result_stretched[0].copy())
al_omega_plus_5.shift(omega_plus_5[0][0].get_center() -
al_omega_plus_5[0][0].get_center())
al_five = omega_plus_5[1].copy()
for bar, dest in zip(al_five, result[1]):
bar.move_to(dest, coor_mask=X_MASK)
al_omega_plus_5.add(al_five)
al_omega_times_3 = VGroup(*result_stretched[:3]).copy()
al_omega_times_3.shift(omega_times_3[0][0].get_center() -
al_omega_times_3[0][0].get_center())
result_in_pow_3 = result_stretched.copy()
result_in_pow_3.shift(omega_pow_3[0][0][0].get_center() -
result_in_pow_3[0][0].get_center())
al_omega_pow_3 = omega_pow_3.copy()
al_omega_pow_3.next_to(result_in_pow_3, buff=0)
al_omega_pow_3[0].add_to_back(*result_in_pow_3.submobjects)
al_ordinals = VGroup(al_omega, al_omega_plus_5, al_omega_times_3,
al_omega_pow_3)
if not fast_play: self.wait_to(17.5)
self.play(ReplacementTransform(ordinals, al_ordinals))
al_ordinals.remove(al_omega_pow_3)
al_ordinals.add(result_in_pow_3)
result_bottom = result.copy()
result_bottom.stretch(0, 1)
result_bottom = VGroup(*[result_bottom.copy() for _ in range(5)])
result_bottom.arrange_submobjects(DOWN, buff=0.3, center=False)
result_bottom.to_edge(DOWN, buff=0)
al_ordinals.add(result_bottom)
mrg_omega = result[0].copy()
mrg_omega_plus_5 = VGroup(result[0], VGroup(*result[1][:5])).copy()
mrg_omega_times_3 = VGroup(*result[:3]).copy()
mrg_ordinals = VGroup(
mrg_omega,
mrg_omega_plus_5,
mrg_omega_times_3,
result.copy(),
VGroup(result.copy()),
)
if not fast_play: self.wait_to(19)
self.play(
FadeOut(dots),
ReplacementTransform(al_ordinals, mrg_ordinals),
)
self.remove(mrg_ordinals)
self.add(result)
if fast_play: return result
brace = BraceDesc(result, "\\omega_1", UP)
self.wait_to(20.2)
self.play(brace.creation_anim())
self.wait_to(22)
conversation = Conversation(self)
conversation.add_bubble(
"Ale podle kapitoly 5 si nemůžeme vzít všechny ordinály.")
self.wait_to(29.5)
conversation.add_bubble("To ne, ale všechny spočetné můžeme.")
self.wait_to(39)
self.add_foreground_mobjects(conversation.dialog)
self.play(FadeOut(VGroup(brace, result)))
omega = OrdinalOmega()
omega.shift(2 * UP)
self.play(ShowCreation(omega))
self.wait_to(48.5)
self.play(FadeOut(conversation.dialog))
def construct(self):
num = 5
ordinals_cnt_base = OrdinalFiniteProd(OrdinalOmega,
num,
x1=-4 + num * 2.5)
ordinals_cnt = VGroup(
[VGroup(ordinals_cnt_base[:i]).copy() for i in range(1, num + 1)])
ordinals_cnt.save_state()
ordinals_cnt.arrange_submobjects(DOWN, center=False, coor_mask=Y_MASK)
ordinals_cnt.to_edge(UP)
self.play(ShowCreation(ordinals_cnt))
#self.wait_to(7)
self.dither()
self.play(ordinals_cnt.restore)
self.remove(ordinals_cnt)
self.add(ordinals_cnt_base)
omega1 = Omega1()
brace = BraceDesc(omega1, "\\omega_1", DOWN)
omega1.line.save_state()
omega1.line.behind_edge(RIGHT)
omega1_bars_src = make_ordinal_power(2)
omega1_bars_src.next_to(ordinals_cnt_base)
omega1_bars_src.add_to_back(*ordinals_cnt_base)
brace.desc.highlight(BLUE)
brace.save_state()
brace.shift_brace(omega1_bars_src)
brace.highlight(BLACK)
self.play(
brace.restore,
omega1.line.restore,
ReplacementTransform(omega1_bars_src, omega1.bars),
)
#self.wait_to(11)
self.dither()
aleph1 = TexMobject("\\aleph_1", "=|", "\omega_1", "|")
aleph1[2].highlight(BLUE)
aleph1[0].highlight(RED)
aleph1.shift(brace.desc.get_center() - aleph1[2].get_center())
aleph1.remove(aleph1[2])
self.play(Write(aleph1))
#self.wait_to(15)
self.dither()
ordinals_a1 = VGroup([omega1.copy() for _ in range(num)])
aleph1[0].save_state()
aleph1[0].next_to(omega1, LEFT, coor_mask=X_MASK)
self.play(
omega1.to_edge,
UP,
FadeOut(VGroup(brace, aleph1[1:])),
MoveFromSaved(aleph1[0]),
)
#self.wait_to(18.5)
self.dither()
aleph1 = aleph1[0]
scale_center = np.array(ordinals_a1[0].line.get_start())
for i, o in enumerate(ordinals_a1):
o.line.scale(1 + i * 0.2, about_point=scale_center)
ordinals_a1_lines_dest = VGroup([o.line for o in ordinals_a1]).copy()
ordinals_a1_bars_dest = VGroup([o.bars for o in ordinals_a1]).copy()
ordinals_a1.arrange_submobjects(DOWN, False, coor_mask=Y_MASK)
ordinals_a1.to_edge(UP)
ordinals_a1_lines = VGroup([o.line for o in ordinals_a1])
ordinals_a1_bars = VGroup([o.bars for o in ordinals_a1])
self.remove(omega1)
self.add(ordinals_a1[0])
self.play(FadeIn(VGroup(ordinals_a1[1:])))
self.play(
Transform(ordinals_a1_lines, ordinals_a1_lines_dest),
Transform(ordinals_a1_bars, ordinals_a1_bars_dest),
)
self.remove(*ordinals_a1[:-1])
omega2_src = ordinals_a1[-1]
omega2 = LongOrdinal(ordinal_end=0.3, line_start=0.2)
brace = BraceDesc(omega2, "\\omega_2", DOWN)
brace.desc.highlight(BLUE)
brace.save_state()
brace.shift_brace(omega2_src)
brace.highlight(BLACK)
self.play(
brace.restore,
ReplacementTransform(omega2_src, omega2),
)
#self.wait_to(24.5)
self.dither()
aleph2 = TexMobject("\\aleph_2", "=|", "\omega_2", "|")
aleph2[2].highlight(BLUE)
aleph2[0].highlight(RED)
aleph2.shift(brace.desc.get_center() - aleph2[2].get_center())
aleph2.remove(aleph2[2])
self.play(Write(aleph2))
#self.wait_to(28)
self.dither()
ordinal_fin = OrdinalFinite(7, x1=SPACE_WIDTH + 1).highlight(BLUE)
ord_labels = VGroup([
TexMobject("\\omega_{}".format(i + 1)).highlight(BLUE).next_to(
bar, UP) for i, bar in enumerate(ordinal_fin[:4])
])
card_labels = VGroup([
TexMobject("\\aleph_{}".format(i + 1)).highlight(RED).next_to(
bar, DOWN) for i, bar in enumerate(ordinal_fin[:4])
])
self.play(
FadeOut(VGroup(aleph2[1:], brace, omega2)),
ReplacementTransform(aleph1, card_labels[0]),
ReplacementTransform(aleph2[0], card_labels[1]),
)
for i, t in enumerate((29, 31, 35.5, 38)):
#self.wait_to(t)
self.play(FadeIn(ord_labels[i]), ShowCreation(ordinal_fin[i]))
if i >= 2: self.play(FadeIn(card_labels[i]))
#self.wait_to(41)
self.play(ShowCreation(VGroup(ordinal_fin[4:], rate_func=rush_into)))
omega = OrdinalOmega().highlight(BLUE)
omega_src = omega.copy()
omega_src.next_to(ordinal_fin, buff=1)
omega_src.add_to_back(*ordinal_fin)
label1 = VGroup(ord_labels[0], card_labels[0])
labels_to_fade = VGroup(list(zip(ord_labels[1:], card_labels[1:])))
labels_to_fade.save_state()
for l, bar in zip(labels_to_fade, omega[1:4]):
l.move_to(bar, coor_mask=X_MASK)
l.highlight(BLACK)
limit_bar = omega[0].copy().next_to(omega)
limit_lord = TexMobject("\\omega_\\omega").highlight(BLUE).next_to(
limit_bar, UP)
limit_lcard = TexMobject("\\aleph_\\omega").highlight(RED).next_to(
limit_bar, DOWN)
limit_g = VGroup(limit_bar, limit_lord, limit_lcard)
limit_g.next_to(omega, coor_mask=X_MASK)
limit_g.save_state()
limit_g.next_to(omega_src, coor_mask=X_MASK)
self.play(
MoveFromSaved(labels_to_fade),
ReplacementTransform(omega_src, omega),
Animation(label1),
limit_g.restore,
)
self.remove(labels_to_fade)
#self.wait_to(52)
plus1_bar = limit_bar.copy()
plus1_label = TexMobject("\\omega_{\\omega+1}").highlight(
BLUE).next_to(plus1_bar, UP)
plus1_g = VGroup(plus1_bar, plus1_label)
plus1_g.next_to(limit_g, coor_mask=X_MASK, buff=0.5)
self.play(FadeIn(plus1_label), ShowCreation(plus1_bar))
#self.wait_to(55)
omega_pow = make_ordinal_power(2, q=(0.8, 0.9, 0.9)).highlight(BLUE)
omega_pow_src = omega_pow.copy()
omega_pow_src.behind_edge(RIGHT)
omega_pow_src[0].add_to_back(limit_bar, plus1_bar)
omega_pow_src.add_to_back(omega)
limit_g.remove(limit_bar)
limit_g.save_state()
plus1_label.save_state()
limit_g.move_to(omega_pow[1][0], coor_mask=X_MASK)
plus1_label.move_to(omega_pow[1][1], coor_mask=X_MASK).highlight(BLACK)
pow_bar = limit_bar.copy()
pow_label = TexMobject("\\omega_{\\omega\\cdot\\omega}").highlight(
BLUE).next_to(pow_bar, UP)
pow_g = VGroup(pow_bar, pow_label)
pow_g.next_to(omega_pow, coor_mask=X_MASK, buff=-.2)
pow_g.save_state()
pow_g.next_to(omega_pow_src, coor_mask=X_MASK)
self.play(
MoveFromSaved(plus1_label),
ReplacementTransform(omega_pow_src, omega_pow),
MoveFromSaved(limit_g),
pow_g.restore,
)
self.remove(plus1_label)
#self.wait_to(59)
self.dither()
labels_to_fade = VGroup(label1, limit_g, pow_label)
labels_to_fade.save_state()
labels_to_fade.highlight(BLACK)
omega1 = Omega1(color=BLUE)
omega1_bar = pow_bar.copy()
omega1_label = TexMobject("\\omega_{\\omega_1}").highlight(
BLUE).next_to(omega1_bar, UP)
omega1_g = VGroup(omega1_bar, omega1_label)
omega1_g.shift(omega1.line.get_end() - omega1_bar.get_center())
omega1_g.add(omega1.line)
omega1_g.save_state()
omega1_g.behind_edge(RIGHT)
limit_g.move_to(omega1.bars[1][0], coor_mask=X_MASK)
pow_label.move_to(omega1.bars[-1], coor_mask=X_MASK)
pow_bar.save_state()
pow_bar.scale(0)
pow_bar.set_stroke(width=0)
pow_bar.move_to(omega1.bars[-1])
self.play(
MoveFromSaved(labels_to_fade, remover=True),
omega1_g.restore,
MoveFromSaved(pow_bar, remover=True),
ReplacementTransform(omega_pow, omega1.bars),
)
#self.wait_to(60+7.5)
self.dither()
ordinal_fin = OrdinalFinite(6, x1=SPACE_WIDTH + 1).highlight(BLUE)
omega1.save_state()
omega1.stretch_about_point(0, 0, ordinal_fin[0].get_center())
labels = VGroup([
TexMobject((i + 1) * "\\omega_{" + "\\omega_1" +
(i + 1) * "}").next_to(bar, UP)
for i, bar in enumerate(ordinal_fin[:3])
])
labels.highlight(BLUE)
next_bar = VGroup(labels[1], ordinal_fin[1])
next_bar.save_state()
next_bar.shift(8 * RIGHT)
self.play(
MoveFromSaved(omega1, remover=True),
next_bar.restore,
ReplacementTransform(omega1_label, labels[0]),
ReplacementTransform(omega1_bar, ordinal_fin[0]),
)
#self.wait_to(60+11)
self.play(FadeIn(labels[2]), ShowCreation(ordinal_fin[2]))
#self.wait_to(60+13)
self.play(ShowCreation(VGroup(ordinal_fin[3:]), rate_func=rush_into))
#self.wait_to(60+21.5)
self.dither(2)
conversation = Conversation(self)
conversation.add_bubble("What about the union of all the ordinals?")
self.play(FadeOut(VGroup(labels, ordinal_fin)),
Animation(conversation.dialog))