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The iterative rule is in the beginning plausible but in the end not dependable, and Zeno is committing each the fallacy of division as well as fallacy of composition.
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Some analysts, one example is Tannery (1887), believe Zeno might have had in mind the paradox was designed to have assumed that both equally House and time are discrete (quantized, atomized) as opposed to continuous, and Zeno intended his argument to challenge the coherence of the idea of discrete Area and time.
Provided an item, we might assume that there is a one, right remedy towards the concern, “What's its position?” Simply because every little thing that exists has a location, and because put by itself exists, so Additionally, it needs to have an area, and so on eternally. That’s a lot of sites, so There's a contradiction. The initial resource is Aristotle’s Physics
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What about Leibniz's infinitesimals or Newton's fluxions? Let us follow infinitesimals, considering that fluxions possess the identical problems and exact resolution. In 1734, Berkeley experienced effectively criticized using infinitesimals as staying "ghosts of departed quantities" that happen to be employed inconsistently in calculus. Earlier Newton had defined instantaneous pace as the ratio of an infinitesimally smaller distance and an infinitesimally small period, and he and Leibniz produced a process of calculating variable speeds that was extremely fruitful. But no person in that century or the following could sufficiently explain what an infinitesimal was. Newton had named them “evanescent divisible portions,” no matter what that intended. Leibniz identified as them “vanishingly little,” but which was just as vague. The practical use of infinitesimals was unsystematic. For instance, the infinitesimal dx is dealt with as remaining equivalent to zero when it is actually declared that x + dx = x, but is taken care of as not getting zero when Utilized in the denominator from the fraction [f(x + dx) - f(x)]/dx and that is the spinoff of the perform f. Also, consider the seemingly evident Archimedean property of pairs of favourable numbers: offered any two good numbers A and B, in case you increase sufficient copies of A, Then you can certainly develop a sum greater than B. This property fails if A is definitely an infinitesimal. Eventually, mathematicians gave up on answering Berkeley’s rates (and so re-described what we imply by common analysis) since, in 1821, Cauchy showed how to accomplish the exact same valuable theorems of calculus through the use of the concept of a limit as more info an alternative to an infinitesimal.
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Spring Break Wife
It was 8am on a Wednesday, and Mom and I were exercising in the basement. I religiously worked out five mornings a week, and she was always right there with me. We started with a run on our two treadmills. Mom's industrial-strength sports bra didn't completely restrain the bounce of her D-cup boobs, which I loved to watch. She didn't seem to notice.
Mom's big tits were a present from Dad for her thirtieth birthday. She told him she didn't want to go through her whole life with tiny A-cup boobs, and he was happy to pay for a boob job. At the time she was also becoming overweight, as thirty-year-old women tend to. So she joined me in my morning workouts. She says I'm her inspiration.
Four years of workouts sculpted Mom into a sleek and strong specimen of feminine perfection with platinum blonde hair. 34D-24-34. 5'6" tall. Long legs. Perky round butt. The fake tits were unnaturally high and firm, not that there's anything wrong with that. She closely resembles April Cheryse, a model. Google her and you'll get the idea.
After running on the treadmills, we lifted free weights then finished with thirty minutes of yoga. The yoga was Mom's idea. I go along with it because it keeps me flexible and it's entertaining to watch her contort herself into all those poses.
When we were done we cooled down for a few minutes on an old sofa in the basement. "So, how are classes going?" I asked.
Mom is a college freshman at age 34. She didn't go to college at the normal age because she had me to take care of. My dad knocked her up when she was only 16 and he was 20. I'm 18 now and a senior in high school.
Dad didn't mind Mom going to college. It amused him in fact. You see, Mom is kind of a bubble-headed blonde. Not exactly the academic type. Neither Dad nor I could picture Mom in a college classroom, but she wanted it so we supported her 100%. She's a great person despite being a bit ditzy.
Mom inherited money from her wealthy family, but Dad is a banker and can afford to support us without dipping into Mom's funds. We had a good, prosperous life in Memphis, Tennessee.
Anyway, back to the story ... Mom's face lit up and she replied, "College is great, Gary. I'm having so much fun!"
"Good. You're fitting in?" I asked. She had been worried about fitting in with the other college freshmen because she was so much older.
Mom giggled. "Yep. I told them I'm 22 and they believe me! They think your old mom is barely older than you! Not bad, huh?"
I wasn't really surprised. She had a young-looking cute face with high cheekbones and dimples, and her body was probably the envy of every girl in her class. "It's because you keep yourself in great shape," I complimented her.
"And I have you to thank for that, honey. You're my inspiration," she replied and kissed me on the cheek. "I wish your dad would work out with us."
Dad was a little older than Mom, 40, and was overweight. As a loan officer at the bank, he spent too much time sitting behind a desk. "Good luck getting him to exercise," I said. Dad was a sedentary kind of guy.
Mom sighed. "I know, but I worry about his health." Then she brightened up and added, "Speaking of your dad, a funny thing happened yesterday."
Mom giggled. "I told Jessica that you're my husband," she said. Jessica was Mom's best friend at college.
"What?! Why would you do that?"
"I didn't mean to. It was an accident," she replied and giggled again. "Jess knows I'm married, and when she saw your photo on my phone, she assumed that you're my husband."
"Why didn't you correct her?" I asked.
Mom looked at me like I was a dunce. "Your dad isn't very believable as the husband of a young college girl, Gary!"
That was true. 40-year-old overweight men with receding hairlines rarely are married to college freshmen. "So you let her think your 18 year old son is your husband?" I asked.
Mom grinned and nodded. "Isn't it brilliant? It really helps me fit in. Jess thinks you're hot, by the way."
"Is she hot? You could set me up on a date with her," I kidded.
Mom frowned. "You're supposed to be my husband, Gary."
"Oh yeah," I chuckled.
Mom seemed to be annoyed with my joke, but a couple minutes later she hesitantly said, "Jessica invited us to go to spring break with her and her boyfriend Brad."
My jaw dropped. "What? Me and you? Mom, I can't go to spring break with you."
She pouted and whined, "Why not? I really really want to go. I won't embarrass you, I promise."
"No, Mom. It's too weird."
"Please?" she begged. "I never got to do stuff like this when I was your age."
She left it unspoken that she didn't do stuff like that because she was taking care of me. I felt the weight of guilt crushing me. Maybe this wasn't too much to ask. Going on a vacation with her wasn't too weird, was it? And it would make her happy. Mom deserved to be happy. "Okay," I agreed. I was sure I'd regret it.
Mom squealed with delight and jumped into my arms. "Thank you! Thank you! Thank you! You're the best son, Gary!" She kissed me on both cheeks and the tip of my nose.
"Don't you mean best husband?" I teased.
She giggled. "Yes. Best spring break hubby." She kissed the tip of my nose again.
"Has Dad agreed to this trip?"
"Not yet. I was hoping you'd help me convince him tonight."
That evening at dinner, Mom told Dad, "Honey, I want to go to spring break with Jessica. She and Brad are going to Fort Lauderdale."
Dad grimaced. "Amy, I don't want to spend my vacation with a bunch of wild college kids."
"I know you don't," Mom answered with a smile. "We'll go on our regular vacation in the summer. Spring break is just for me."
Dad's eyebrows raised. "You want to go to spring break alone?"
Mom shook her head. "Not alone. With my friend Jessica and her boyfriend Brad. And with Gary."
"Gary?" Dad looked at me. "Do you want to go to spring break?" He thought about his question for just a moment, then said, "Stupid question. All 18-year-old boys want to go to spring break." Dad looked back and forth between me and Mom for a minute then said, "Alright. I don't see any reason why not. The two of you can take care of each other and make sure you don't get into trouble."
Mom squealed, jumped up and hugged Dad. "Thank you, Mark! Gary and I will take good care of each other."
She didn't tell him that her friends thought I was her husband, and I sure wasn't going to say anything. It was too strange to mention.
The morning we were to leave for spring break, Mom kissed Dad goodbye at the door. He was going to play golf with his friends. "Thanks again for letting me go," Mom said.
"You're welcome. Have fun but don't drink too much and don't do drugs," he urged.
"We won't," she assured him. One more kiss and he left. She closed the door and turned to me. "Ready for a fun week, Gary?"
"Yeah, Mom. This should be a blast."
She hugged me and looked up into my eyes. I was about 4 inches taller. "No more calling me 'Mom', dummy," she scolded. "I'm your wife this week, remember? You have to call me Amy. Say it. Amy."
"Good," she replied, then kissed me on the lips.
I jerked back from her. "What was that?!" I yelped.
"Practice," she told me. "And you failed. We're supposed to be married. If you act like that when we kiss, Jessica will know something is up!"
"Why do we have to kiss?"
"We're married. Married couples kiss. Particularly young married couples. Let's try it again." She planted another kiss on my lips. I held myself rigid for a minute, then finally gave in and returned the kiss a little.
Mom broke the kiss and pouted. "You didn't freak out that time, but it still wasn't good, Gary. This is important to me, honey. Please don't ruin spring break for me. Try a little harder, okay?" she pleaded.
She knew I'd do anything for her when she did that puppy dog eye thing. I dipped her and gave her a huge dramatic smooch. "Better?" I asked.
Mom came up laughing. "Maybe a little too much." She hugged me tight. "Thank you for helping me fit in with my college friends, honey."
"No problem." She gave me another quick peck on the lips before Jessica and Brad pulled into our driveway.
Jessica was a cute little brunette, shorter than Mom and kind of chunky. But she had a bubbly personality just like Mom's and I could see why they were friends. Brad was a big strong dude. A couple inches taller than me and heavier with muscles. He was a cool guy and we got along well right away.
"Do y'all own that house?" Jessica asked, staring at our impressive family home. "How can you afford it?"
"I have an inheritance from my grandfather," Mom explained.
"Wow," Jessica marveled. "So you bought this place for you and Gary?"
"Yep," Mom replied. "As soon as Gary turned 18, we married and I moved him in with me. His mom wasn't too happy about that. She's kind of a bitch." She winked at me.
"So you're married and have a house and you're still in high school?" Brad asked me.
"Yeah, I graduate in May," I answered.
"Cool," he responded with admiration.
It was Brad's SUV and he felt most comfortable driving. So he drove for 3 hours, then took a break while Jess drove for an hour or so. Then he drove again. It was fine with me. I think driving is boring.
I dozed off on the back seat, and when I woke Mom was stretched out on the seat with her head on my lap. Jessica noticed that I was awake and softly said, "You two are so cute together. It's obvious how much she loves you."
I blushed. "Thanks. I really love her too."
Mom must have been awake because she cheerfully said, "That's so sweet, honey." She reached up and pulled my head down to hers. Her lips met mine in a scorching kiss that lasted for at least a minute and left me breathless.
Jessica giggled. "I think you dazed him with that one, Amy."
Mom giggled too and scooted up to sit on my lap. "That's the way I like him best, dazed and confused, and I work hard to keep him that way," she teased. Her head rested on my shoulder.
A pretty girl on your lap always feels good, even if it's your mom. I loosely wrapped my arms around her waist and we snuggled.
"Aww, so cute," Jess cooed from the front seat.
It was close to midnight when we finally made it to Fort Lauderdale. When we checked in, I was stunned to learn that we all were sharing one room. "We don't have our own room?" I asked Mom.
"No, silly. That's the whole reason Jessica invited us. To split the cost of the room so she and Brad could afford to come. There's nothing wrong with a man and wife sharing a room with another couple," she dismissed my concerns.
My eyes boggled. "You don't mind sharing a bed with me?" I whispered.
Mom looked at me puzzled, then reached out and tousled my hair. "Stop being silly. This week we're married, so it's fine. Now get with the program, Gary. If you don't start acting like a loving husband, Jessica and Brad will know something is wrong."
While Jessica and Brad were occupied, Mom stepped outside to call home and let Dad know we arrived safely.
We went to the room, and I tried to act natural. We were all exhausted so we got ready for bed. Mom wore an oversize t-shirt to bed, and I wore a t-shirt and boxer shorts.
We got into our beds and turned out the light. It wasn't long before I heard noises from the other bed. I guess Brad and Jessica couldn't keep their hands off each other, even though we we right there. Brad huffed like a locomotive and Jessica made weird little squeaking noises.
Mom and I both turned on our sides, with our backs to Jessica and Brad. "Eep, eep, eep!" Jessica's funny little squeaks continued as Brad fucked her.
Mom started giggling at the crazy situation, and then Jessica started giggling too. "I can't help it," the blonde laughed, still in the midst of fucking. "Those noises just come out." That made Brad laugh, and then me too. It removed the tension from the room.
When they were done fucking, Jessica said, "Sorry for embarrassing you guys, but I really needed that."
"You don't have apologize, Jess," Mom replied. "You're on vacation and we're all adults. Have fun."
"Thanks, Amy. If you and Gary want to do it too, go right ahead," she said with a giggle.
"I think we'll just sleep, Jess. Goodnight."
When I woke I was spooned against Mom's back. We were still on our sides, but I must have snuggled up to her in the night. My arm was draped over her stomach, so I lifted it in embarrassment. Mom grabbed it and put it back around her. "Don't move," she twisted her head to me and whispered. "It's comforting to be with you like this. Did you sleep well?"
"I slept great," I whispered. And I did. I slept through the night like a log.
Mom smiled. "Me too."
Brad and Jessica got up and went into the bathroom to shower together. Mom rolled over to face me. "This is a lot less awkward than I thought it would be," Mom said softly. "I like cuddling with my boy."
"I like it too," I confided. I struggled to keep my morning hard-on from pressing against Mom's perfect ass.
When Brad and Jessica were done in the bathroom, Mom told me to use it next. I shat, showered, shaved and put on my red swim trunks.
Mom used the bathroom last, and came out wearing a new bikini. "What do you think?" she asked and wagged her hips in a saucy way at me.
"Jesus, Amy! That's hot!" I exclaimed. Small triangles held her round fake tits and the bikini bottom was cut very low. She turned around and I saw that the back only covered a fraction of her tight little butt.
"It is hot," Jessica agreed.
Brad put his eyes back in his head and rolled his tongue back into his mouth. "Yeah. It's hot," he simply said.
Both girls giggled and Mom hugged me. The beach was full of pretty college girls, but in that bikini Mom attracted a ton of attention. Guys eyed her with lust as she sauntered around until we found an empty patch of sand for our blankets.
A big group of college kids were having a splash battle in the ocean. The four of us quickly joined in.
The splash battle intensified when Jessica and another girl jumped on Brad and pushed him underwater. He came up sputtering and everyone laughed. In retaliation, several guys tackled their girlfriends and dunked them.
I advanced on Mom and she looked worried, but still smiling. "Gary, don't," she begged and laughed as she tried to back away. I jumped, cleanly knocked her off her feet and dunked her with a big splash.
I helped her back to her feet and kept my hands on her waist to steady her. She pushed her wet hair out of her face. "Sorry," I apologized. "I couldn't resist."
"It's okay. This is fun," she replied with a smile. She leaned against me to whisper in my ear, "I'm glad you're here, Gary. I couldn't relax and enjoy myself with anyone but you."
"I'm glad too," I replied. I tried not to perv out on the feel of here her soft tits compressed against my bare chest, or the sight of water dripping down her face. There's just something about a soaking wet girl that revs my engine.
The splash battle wound down and the others were headed for the beach. I let go of Mom and we followed.
We spent most of that day doing the usual beach stuff. Sunning, reading, trying to ride the little waves, throwing a frisbee, taking walks. Mom played and had as much fun as any of the young college kids. She really did fit in with them. It made me proud of her.
The girls wanted to get dressed up that night. Brad and I just wore khakis and polo shirts, but we learned the girls shopped together before the trip. They had matching little black dresses. They were sleeveless and the neckline dropped low, displaying the perfect cleavage between Mom's fake tits. The dresses were tight and short too, ending only a few inches down their thighs. Very sexy. I whistled. "Wow. You ladies are gorgeous," I complimented.
"Yeah babes, you look great," Brad agreed. Jessica gave him a kiss, so Mom smirked at me and planted one on my lips too. I didn't flinch or anything. She's a good kisser.
Mom happily took my arm and we went to a nice restaurant for dinner. We went to a nightclub after.
The club was an upscale one. They played "American standards" music, like Sinatra, and it was all slow-dancing. Brad and Jessica went right on the floor to dance, but I wasn't confident in my ability so I didn't ask Mom. She kept looking at me, expecting me to ask her, but I just watched the others.
A big heavy guy who looked like an offensive lineman came right up to Mom and said, "Come on, babe. Let's dance." He took her hand and pulled her toward the dance floor. She looked back at me but I didn't know what to do. I didn't want to stop her from dancing and having a good time if she wanted to.
The guy drew her into his arms and danced them into the crowd. I tried to keep an eye on her but the other dancers got in the way. Two, then three songs passed and Mom didn't return. I started to get mad. Mad at myself for not asking her to dance and at her for leaving me here alone and dancing with that guy.
Mom came back into view and I saw that the asshole had a big hand on her ass. I rushed out to her. "That's enough," I told the guy.
He sneered at me. "Beat it, wimp. The girl is with me."
"If you don't take your hands off my wife, I'm going to break your fucking nose," I threatened.
The asshole looked like he wouldn't mind a fight, but Mom told him, "Thank you for the dances, Frank, but my husband will take over now." To his credit he backed away without a fight.
Mom had a little smirk on her lips that annoyed me. I gathered her in my arms and danced with her. She leaned in, pressing the full length of her curvy body against mine, and whispered in my ear, "What took you so long?"
"What?" I asked, confused and still mad.
"What took you so long to get me away from that guy? I'm your wife this week, honey. You shouldn't let other men dance with me."
I stared in her eyes, but couldn't figure out if she was teasing me. "I thought you wanted to dance with him."
She shrugged but her eyes sparkled merrily. She was enjoying this but I had no idea what was going on. "I wanted to dance with you, but he asked and you didn't," she explained. "And you know how much I like to dance. You shouldn't have let me go with him."
She shook her head. "I'm your wife. You need to assert yourself. He had his hands all over me." She was still smirking.
"I know," I groused. "I can't believe you let him do that."
"I expected you to stop him," she replied innocently. "You're my big strong husband. You're supposed to protect me."
I blushed, ashamed by my inaction. "Sorry," I apologized.
She giggled and nipped my earlobe with her teeth. "It's okay. I think you learned your lesson."
We stayed at the nightclub for hours. The music wasn't my favorite, but I have to admit that slow dancing was a hell of a lot of fun after Mom taught me some basic steps. Mom stayed in my arms the rest of the night, mostly swaying on the dance floor but also with my arm around her while at our table.
I used the restroom once, and damned if some guy wasn't chatting up Mom when I got back. I possessively put my arm around her waist and glared at the guy. Mom kissed my cheek and told him, "This is my husband and we're going to dance again."
Out on the dancefloor I complained, "What the hell, Mom? I'm gone for a minute and you're flirting with some guy again."
She gave me that smirk. "I wanted to see what you'd do, and you passed with flying colors, darling." She kissed me gently. "Don't worry, I wasn't going to dance with him."
"It didn't look that way to me." Seeing her flirting with men really pissed me off. She was married and my mom!
"Don't sulk, Gary," she chastised. "If you're going to act like a petulant little boy then you should go back to the hotel and I'll find a real man to dance with. It would be easy. Half the men in here are eying me like a piece of meat."